Continuous and discrete higher-degree F-transforms based on B-splines

Abstract

The paper deals with the continuous and discrete higher-degree fuzzy transforms (F-transforms with polynomial components) with respect to a generalized fuzzy partition given by B-splines. We investigate properties of the direct and inverse F-transforms in these cases and prove that using B-splines allows us to improve the quality of approximation of smooth functions.

1 Introduction

We assume that the reader is familiar with the concept of fuzzy transform (F-transform or $F^0$-transform) introduced in Perfilieva and Haldeeva (2001) (see also the key paper Perfilieva 2006) and generalized to the case of higher degree by Perfilieva et al. (2011). As it was shown in Perfilieva et al. (2011), the ordinary F-transform with constant components can be extended to the F-transform with m-degree polynomial components, $m \ge 0$ (the $F^{m}$-transform). Theory and applications of $F^{m}$-transforms in image processing, data analysis and time series analysis have been rapidly developed during last years. The technique of the F-transform of a higher degree has been generalized in many directions: discrete $F^{m}$-transforms, $F^{m}$-transforms of functions of many variables, as well as $F^{m}$-transforms with respect to a generalized fuzzy partition.

The key elements of the F-transform are basic functions which form a fuzzy partition. A generalized fuzzy partition for the ordinary F-transform has been introduced by Stefanini (see Stefanini 2009, 2011). It was fuzzy k-partition with basic functions of bandwidth k (i.e., basic functions, whose support consists of 2k intervals). The result of such generalization is a denser fuzzy partition that may improve the approximation properties and the smoothness of the inverse F-transform. The concept of fuzzy k-partitions has been extended by I. Perfilieva with co-authors (see, e.g., Perfilieva 2015; Holčapek et al. 2015) and applied to the higher degree case of F-transform.

There are a number of papers dealing with fuzzy transforms with respect to a fuzzy partition with specially designed basic functions. In this paper we investigate the $F^{m}$-transform, $m\ge 0$, with respect to a generalized fuzzy partition based on central, odd-degree B-splines. The idea of using splines for forming a fuzzy partition is not new (see, e.g., Bede and Rudas 2011; Kodorane and Asmuss 2013). Bede and Rudas (2011) had considered the case when a fuzzy partition is given by means of B-splines, Kodorane and Asmuss (2013) had suggested an alternative approach and introduced a new type of fuzzy partitions based on splines. In both cases the ordinary F-transform has been considered.

Our research focuses on approximation properties of the $F^m$-transform with respect to a generalized uniform fuzzy partition given by B-splines of degree $2k-1$. The first result on approximation properties of the continuous $F^m$-transform in this case was proposed by the authors at the FUZZ-IEEE 2015 conference (Kokainis and Asmuss 2015). A discrete version of this result was announced at the FSTA 2016 conference. In this paper we prove that the continuous and discrete $F^m$-transforms based on B-splines of degree $2k-1$ are precise for polynomials of degree $r\le \min \left\{ 2m+1,2k-1 \right\}$. On the basis of this result we obtain error estimations for approximation of a transformed function and its derivatives by the continuous and discrete $F^m$-transform and prove that using B-splines may improve the approximation properties of the technique of higher-degree F-transform.

The paper is organized as follows. Section 2 contains preliminaries on $F^m$-transforms (direct and inverse, continuous and discrete) and B-splines. Section 3 introduces odd-degree B-splines as a special tool of fuzzy partitions. Sections 4 and 5 are devoted to properties of the continuous and discrete $F^m$-transforms based on B-splines. Section 7 identifies the degree of polynomials for which the continuous and discrete $F^m$-transforms based on B-splines are precise. Section 8 presents approximation properties of the B-spline based $F^m$-transforms. Section 9 contains proofs of the technical claims of the previous sections. Finally, Sect. 10 concludes the obtained results.

2 Preliminaries

Here we remind the concept of higher-degree direct F-transform and the corresponding inverse transform. In Sect. 2.6, key facts about B-splines are recalled.

Let $\mathbb {N}$ be the set of positive integers and $\mathbb {N}_0 = \mathbb {N} \cup \left\{ 0 \right\}$ be the set of nonnegative integers. The set $\left\{ n,n+1,\ldots , m \right\}$, for integers n, m with $n \le m$, is denoted by $[n \, ..\, m]$. The space of univariate polynomials of degree at most l is denoted by $\mathbb {P}_l$.

2.1 Generalized fuzzy partition

Suppose that $[a,b] \subset \mathbb {R}$, $a<b$, and $N \in \mathbb {N}$ is chosen. Let $t_0, \ldots , t_N$ be fixed nodes s.t. $a \le t_0< \ldots < t_N \le b$. For each $i \in [0 \, ..\, N]$ nonnegative reals $h_i'$, $h_i''$ are chosen which fulfill the following requirements:

• $h_i' + h_i'' > 0$;

• $\bar{E}_i \subset [a,b]$, where we denote

  $$\begin{aligned} E_i := (t_i-h_i', t_i + h_i''), \quad \bar{E}_i := [t_i-h_i', t_i + h_i'']; \end{aligned}$$

• $\bigcup _{i=0}^N \bar{E}_i = [ a, b]$.

Definition 1

Fuzzy sets $A_0, \ldots , A_N : [a,b] \rightarrow [0,1]$ are said to constitute a generalized fuzzy partition $\mathbb {A}$ (with nodes $t_i$ and margins $h_i', h_i''$, $i \in [0 \, ..\, N]$) of [a, b] if the following conditions are met for all $i\in [0 \, ..\, N]$:

• (locality): $A_i(t) > 0$ if $t \in E_i$ and $A_i(t) = 0$ if $t \in [ a, b] \setminus E_i$;

• (continuity): $A_i$ is continuous on $\bar{E}_i$;

• (covering): $\sum _{j=0}^{N} A_j(t) > 0$ for all $t \in (a,b)$.

Remark: note a minor difference of the definition from Perfilieva and Kreinovich (2013)—originally covering condition required the strict inequality for $t \in \left[ a,b \right]$.

We work with a uniform partition in the following sense:

Definition 2

Let $h >0$ and $h' > h/2$. Then the partition $\mathbb {A}$ is said to be $(h,h')$-uniform, if the following properties are satisfied:

• $t_{i+1} = t_i + h \$ for all $i \in [0 \, ..\, N-1]$, and $h_i' = h_i'' = h'$ for each $i \in [0 \, ..\, N]$;

• $A_i(t_i - t) = A_i(t_i + t)$ for all $t \in [0,h']$, ${i\in [0 \, ..\, N]}$;

• $A_{i}(t) = A_{i+1}(t+h)$ for all $t \in \bar{E}_i$ and $i\in [0 \, ..\, N-1]$.

In this case $E_i = (t_i - h', t_i + h')$.

It is easy to see that then there is an even function ${A : [- H,H] \rightarrow \mathbb {R} }$, $H:=h'/h$, s.t. for all $i\in [0 \, ..\, N]$ and $t \in \bar{E}_i$ it holds that $A_i(t) = A\left( \frac{t-t_i}{h} \right)$. The function A is said to be the generating function of the partition $\mathbb {A}$.

Throughout the rest of this section, let an interval $[a,b] \subset \mathbb {R}$, $a<b$, and its generalized fuzzy partition $\mathbb {A}= \left\{ A_0, \ldots , A_N \right\}$ be fixed. Fix also an integer $m \ge 0$.

2.2 Integral direct $F ^m$-transforms

For each $i \in [0 \, ..\, N]$ denote by $L_2^{}(A_{i})$ the Hilbert space of all square integrable functions ${f : \bar{E}_i \rightarrow \mathbb {R}}$ with the inner product $\left\langle \cdot ,\cdot \right\rangle _{i}$ (and the induced norm $\left\| f \right\| _i := \left\langle f,f \right\rangle _{i}^{0.5}$) given by

$$\begin{aligned} \left\langle f,g \right\rangle _{i} = \int \limits _{\bar{E}_i} f(t)g(t) A_i(t) \,\mathrm {d}t. \end{aligned}$$

(1)

Remark: in fact, this way a semi-norm $\left\| \cdot \right\| _i$ in the space $L_2^{}(A_{i})$ is defined. Identifying functions f and g, which are almost everywhere equal in the sense $\left\| f-g \right\| _i = 0$, turns $\left\| \cdot \right\| _i$ from a semi-norm into a norm and the semi-definite bilinear form $\left\langle \cdot ,\cdot \right\rangle _{i}$ into an inner product. Further on, this technical detail will be ignored.

The set of all functions $f : [ a, b] \rightarrow \mathbb {R}$, s.t. for all $i \in [0 \, ..\, N]$, $f |_{\bar{E}_i} \in L_2^{}(A_{i})$, is denoted by $L_2(\mathbb {A})$.

Fix $i \in [0 \, ..\, N]$. The space $L_2^m(A_i)$ (of dimension $m+1$) is defined as the closed linear subspace of $L_2^{}(A_{i})$, spanned by the set of linearly independent system of polynomials $\left\{ 1, t, t^2, \ldots , t^m \right\}$ (restricted to $\bar{E}_i$).

As in Perfilieva et al. (2011), an orthogonal basis $\{\mathtt {P}_{i,0}, \ldots , \mathtt {P}_{i,m}\}$ of $L_2^m(A_i)$ can be constructed so that $\deg \mathtt {P}_{i,l} = l$, $l \in [0 \, ..\, m]$, and the leading coefficient of $\mathtt {P}_{i,l}$ is equal to 1.

Definition 3

Let $f \in L_2( \mathbb {A})$. By $F_{m,i}^\rightarrow [f]$ the orthogonal projection of $f |_{\bar{E}_i}$ on $L_2^m(A_i)$ is denoted, $i \in [0 \, ..\, N]$. The vector ${F}_m^\rightarrow [f]= ( F_{m,0}^\rightarrow , \ldots , F_{m,N}^\rightarrow )$ is said to be the direct $F^m$-transform of f w.r.t. the fuzzy partition $\mathbb {A}$.

One can see that the components of the direct $F^m$-transform of f can be computed as

$$\begin{aligned} F_{m,i}^\rightarrow [f](t) = \sum _{l=0}^m c_{i,l}[f] \mathtt {P}_{i,l}(t) , \quad i \in [0 \, ..\, N], \end{aligned}$$

where

$$\begin{aligned} c_{i,l}[f] := \frac{\left\langle f,\mathtt {P}_{i,l} \right\rangle _{i}}{\left\| \mathtt {P}_{i,l} \right\| _{i}^2} ,\quad l \in [0 \, ..\, m], \ i \in [0 \, ..\, N] . \end{aligned}$$

(2)

If no confusion can appear, this notation is simplified to $c_{i,l}$.

Lemma 1

Suppose that $f \in \mathbb {P}_{r}$, for some $r \in \mathbb {N}_0$, is expressed as

$$\begin{aligned} f = \sum _{l=0}^{r} \alpha _{l} \mathtt {P}_{i,l} \end{aligned}$$

for some $i \in [0 \, ..\, N]$ and real coefficients $\alpha _{l}$. Denote $\alpha _l = 0$ for all $l > r$.

Then $c_{i,l} = \alpha _l$ for all $l \in [0 \, ..\, m]$ and, consequently, the ith component of the direct $F^m$-transform equals

$$\begin{aligned} F_{m,i}^\rightarrow [f](t ) = \sum _{l=0}^m \alpha _{l} \mathtt {P}_{i,l} (t). \end{aligned}$$

Proof

From the orthogonality of $\mathtt {P}_{i,l}$, $l \in [0 \, ..\, m]$, it follows that

$$\begin{aligned} \left\langle f,\mathtt {P}_{i,l} \right\rangle _{i} = \alpha _{l}\left\| \mathtt {P}_{i,l} \right\| _{i}^2 . \end{aligned}$$

Since $\left\| \mathtt {P}_{i,l} \right\| _{i} \ne 0$, this implies $\alpha _l = c_{i,l}$. $\square$

2.3 Discrete direct $F ^m$-transform

Let $\varDelta = \left\{ z_{1},\ldots ,z_{L} \right\} \subset [a,b]$ be a discrete set of the interval [a, b] and $f : \varDelta \rightarrow \mathbb {R}$. Denote $y_j = f(z_{j})$, for each $j \in [1\, ..\, L]$, and let $\mathbf {Y} = \left( y_1, y_2,\ldots ,y_{L} \right) ^T$ be a column vector containing the values of the function f.

For each $i \in [0 \, ..\, N]$ construct matrices

$$\begin{aligned}&\mathbf {X}_i = \begin{pmatrix} 1 &{} z_{1} - t_i &{} (z_{1} - t_i)^2 &{} \ldots &{} (z_{1} - t_i)^m\\ &{} \ldots \\ 1 &{} z_{L} - t_i &{} (z_{L} - t_i)^2 &{} \ldots &{} (z_{L} - t_i)^m \end{pmatrix} \\&\mathbf {A}_i = {{\mathrm{diag}}}(A_i(z_{1}), A_i(z_{2}), \ldots , A_i (z_{L}) ). \end{aligned}$$

Definition 4

We say that the set $\varDelta$, i.e., the domain of f, is sufficiently dense in the fuzzy partition $\mathbb {A}$ w.r.t. m if the matrix $\mathbf {X}_i^T \mathbf {A}_i \mathbf {X}_i$ is invertible for each $i \in [0 \, ..\, N]$.

Notice that this definition slightly differs from Holčapek and Tichý (2014, Def. 3).

Finally, if the set $\varDelta$ is sufficiently dense (in the fuzzy partition $\mathbb {A}$ w.r.t. m), then the discrete direct $F^m$-transform can be defined:

Definition 5

(Holčapek and Tichý 2014) The vector ${F}_m^\rightarrow [f]$ is the discrete direct $F^m$-transform of function f w.r.t. the fuzzy partition $\mathbb {A}$, if the ith component $F_{m,i}^\rightarrow [f]$, $i\in [0 \, ..\, N]$, of this vector is the polynomial

$$\begin{aligned} F_{m,i}^\rightarrow [f] (t) = \sum _{j=0}^{m} \beta ^{(i)}_j (t - t_i)^j, \quad t\in \mathbb {R}, \end{aligned}$$

where

$$\begin{aligned} \beta ^{(i)}= \left( \mathbf {X}_i^T \mathbf {A}_i \mathbf {X}_i \right) ^{-1} \mathbf {X}_i^T \mathbf {A}_i \mathbf {Y}. \end{aligned}$$

2.4 Inverse and composite $F ^m$-transforms

Definition 6

Let $\mathbf {p} = \left( p_0,\ldots , p_N \right) \in \mathbb {P}^{N+1}_m$ be a tuple of $N+1$ polynomials of degree at most m. Then the function

$$\begin{aligned} {F}_m^\leftarrow [\mathbf {p}](t) = \frac{\sum _{i=0}^N p_i(t) \, A_i(t)}{\sum _{i=0}^N A_i(t)}, \quad t \in (a,b), \end{aligned}$$

is called the inverse $F^m$-transform of $\mathbf {p}$ w.r.t. the fuzzy partition $\mathbb {A}$.

One can see that the inverse $F^m$-transform is applied to a tuple of polynomials and is defined as the linear combination of these polynomials with weight functions. In such framework the inverse fuzzy transform can be considered irrespective of the direct one. If our goal is to obtain an approximation of a function f, we need to apply the inverse $F^m$-transform ${F}_m^\leftarrow$ to the result ${F}_m^\rightarrow [f]$ of the direct $F^m$-transform, i.e., we need to take the composition of two transforms.

Definition 7

Let $f : D_f \rightarrow \mathbb {R}$, where the set $D_f \subset [a,b]$ is

• either the whole interval [a, b] (in the case of the integral $F^m$-transform; then we also require f to be from the space $L_2( \mathbb {A})$),

• or the discrete grid $\varDelta$, described in Sect. 2.3 (in the case of the discrete $F^m$-transform; then we also require that its discrete direct $F^m$-transform ${F}_m^\rightarrow [f]$ exists).

Suppose that the direct $F^m$-transform (either the integral or the discrete version) of f w.r.t. the fuzzy partition $\mathbb {A}$ is

$$\begin{aligned} {F}_m^\rightarrow [f] = ( F_{m,0}^\rightarrow [f], \ldots , F_{m,N}^\rightarrow [f]) \in \mathbb {P}^{N+1}_m . \end{aligned}$$

Then the inverse $F^m$-transform of this vector, i.e., the function

$$\begin{aligned} {F}_m[f](t) = \frac{\sum _{i=0}^N F_{m,i}^\rightarrow {}[f](t) \, A_i(t)}{\sum _{i=0}^N A_i(t)}, \quad t \in (a,b), \end{aligned}$$

is called the (composite) $F^m$ -transform of f w.r.t. the fuzzy partition $\mathbb {A}$.

It can be seen that the direct $F^m$-transform is a mapping from a space of functions to the space $\mathbb {P}^{N+1}_m$, whereas the inverse $F^m$-transform maps a vector of polynomials from $\mathbb {P}^{N+1}_m$ back to a function. Hence, the composite transform is a composition of these two mappings. We will often omit “composite” and simply say “$F^m$-transform of f”.

Remark: in the original definition of $F^m$-transform this composition is considered to be the inverse $F^m$-transform, i.e., originally the inverse $F^m$-transform is applied to a transformed function f, not to the result of its direct $F^m$-transform.

2.5 Ruspini condition

Suppose that $[\hat{a}, \hat{b}] \subset [a,b]$, $\hat{a} < \hat{b}$, is such an interval that

$$\begin{aligned} \sum _{i=0}^{N} A_i(t) = 1 \quad \text {for all } t \in [\hat{a}, \hat{b}] . \end{aligned}$$

In this case we say that the generalized fuzzy partition $\mathbb {A}$ fulfills the Ruspini condition on $[\hat{a}, \hat{b}]$. It is obvious that in this interval the expression of the inverse $F^m$-transform of a tuple $\mathbf {p} = \left( p_0,\ldots , p_N \right) \in \mathbb {P}^{N+1}_m$ can be simplified to

$$\begin{aligned} {F}_m^\leftarrow [\mathbf {p}](t) = \sum _{i=0}^N p_i(t) \, A_i(t), \quad t \in [\hat{a}, \hat{b}]. \end{aligned}$$

Also the expression of the composite $F^m$-transform can be simplified accordingly:

$$\begin{aligned} {F}_m[f](t) = \sum _{i=0}^N F_{m,i}^\rightarrow {}[f](t) \, A_i(t) ,\quad t \in [\hat{a}, \hat{b}]. \end{aligned}$$

The following statement can be proven analogously to Lemma 8 of Perfilieva et al. (2011):

Lemma 2

For all $p \in \mathbb {P}_m$ the $F^m$-transform of p coincides with p on $[\hat{a}, \hat{b}]$ (the interval where the generalized fuzzy partition fulfills the Ruspini condition), i.e., $p(t) = {F}_m[p](t)$ for all $t \in [\hat{a}, \hat{b}]$.

2.6 B-splines

The way we define B-splines here follows the definition from Schoenberg (1973) (see also Schoenberg 1964).

2.6.1 General definition and properties

The truncated power function $t^n_+$ with integer $n \ge 0$ is defined as follows:

$$\begin{aligned} t^n_+ = {\left\{ \begin{array}{ll} t^n , &{} t > 0;\\ 0, &{} t \le 0. \end{array}\right. } \end{aligned}$$

Let

$$\begin{aligned} \psi _n (t,\tau ) = (-1)^{n+1} \, (n+1) \, (t-\tau )_+^n. \end{aligned}$$

Suppose that $-\infty< t_i< t_{i+1}< \ldots< t_{i+n+1} < + \infty$. Then the function $B^n_i$, defined as the divided difference of order $n+1$ of $\psi _n(t,\tau )$ with respect to the variable $\tau$ and based on $n+2$ points $t_i, t_{i+1}, \ldots , t_{i+n+1}$, is called a B-spline (of degree n with knots $t_i$, $t_{i+1}, \ldots , t_{i+n+1}$).

By $\phi _n$ we denote the central B-spline (Schoenberg 1973), which is obtained when $t_i = -\frac{n+1}{2}$ and $t_{i+j} = t_i + j , j \in [0 \, ..\, n+1]$.

The functions $B^n_i$ are piecewise polynomial functions satisfying

• the restriction of $B^n_i$ to $(t_{i+j},t_{i+j+1} )$, $j\in [0 \, ..\, N]$, is a polynomial;

• $B^n_i \in C^{n-1}(\mathbb {R})$;

• $B^n_i(t) > 0$ if $t \in \left( t_i, t_{i+n+1} \right)$;

• $B^n_i(t) = 0$ if $t \notin \left( t_i, t_{i+n+1} \right)$.

B-splines can be constructed through the following recurrence relation (Boor 2001):

$$\begin{aligned}&\frac{n}{n+1} B_i^n (t) \nonumber \\&\quad = \frac{t - t_i}{t_{i+n+1} - t_i}B_i^{n-1} (t) + \frac{t_{i+n+1} - t}{t_{i+n+1} - t_i}B_{i+1}^{n-1} (t) \end{aligned}$$

(3)

for $n \in \mathbb {N}$ and $t \in \mathbb {R}$, noting that $B^0_i (t ) = 1/(t_{i+1} - t_i)$ for all $t \in \left( t_i, t_{i+ 1} \right)$. Moreover, the central B-spline $\phi _n$ satisfies the following properties:

• $\phi _n$ is an even function;

• $\phi _n$ is non-decreasing on $(-\infty ;0)$ and non-increasing on $(0;+\infty )$;

• when $n=2k-1$ is an odd number, $\phi _{2k-1}$ has integer knots $-k$, ..., k.

2.6.2 Marsden’s identity

Suppose that $i + n \le j$ and there is a sequence of knots

$$\begin{aligned} -\infty< t_i< t_{i+1}< \cdots< t_{j}< \cdots< t_{j+n+1} <+ \infty . \end{aligned}$$

Then, for all $z \in \mathbb {R}$ and x satisfying $t_{i+n} < x \le t_{j+1}$, the following equality holds (Marsden 1970):

$$\begin{aligned} (z-x)^n= \sum _{l=i}^{j} \frac{t_{l+n+1} - t_l}{n+1} \, \prod \limits _{s = l+1}^{l+n} (z-t_s) B^n_l(x). \end{aligned}$$

(4)

An easy consequence (Marsden 1970) is the equality

$$\begin{aligned} \sum _{l=i}^{j} \frac{t_{l+n+1} - t_l}{n+1} \, B^n_l(x) = 1, \end{aligned}$$

which is valid for $x \in (t_{i+n}, t_{j+1})$.

In particular, when the knots are 1-equidistant, i.e., $t_{i+l+1} =t_{i+l} +1$ for all $l \in [0 \, ..\, j-i]$, the equality $B^n_{i+l}(x) = B^n_i(x -l )$ holds and we obtain

$$\begin{aligned} \sum _{l=0}^{j-i} B^n_i(x -l ) = 1 \quad \text { for all }x \in (t_{i+n}, t_{j+1}). \end{aligned}$$

(5)

If $n \ge 1$, then this is true also for $x = t_{i+n}$ and $x=t_{j+1}$.

3 Generalized fuzzy partition based on the central B-spline of odd degree

3.1 Construction of the central B-splines of odd degree

It can be easily seen from (3) that the central B-spline satisfies the following recurrence relation for all $n \in \mathbb {N}$:

$$\begin{aligned} \phi _n (t) = w_{t}^{n,0} \, \phi _{n-1} (t+0.5) + w_{t}^{n,1} \, \phi _{n-1} (t-0.5) \end{aligned}$$

(6)

for all $t \in \mathbb {R}$, where

$$\begin{aligned} w_{t}^{n,0} :=\frac{ n+1 + 2 t}{2n} , \quad w_{t}^{n,1} := \frac{ n+1 - 2 t}{2n} , \end{aligned}$$

and $\phi _0(t) = 1$, $t \in (-0.5; \, 0.5)$. For example, it can be calculated that

$$\begin{aligned} \phi _1(t) = {\left\{ \begin{array}{ll} 1 - \left| t \right| , &{} t \in (-1,1);\\ 0 , &{} t \in \mathbb {R}\setminus (-1;1). \end{array}\right. } \end{aligned}$$

(7)

Let $k \in \mathbb {N}$. As it was mentioned previously, the central B-spline $\phi _{2k-1} \in C^{2k-2}(\mathbb {R})$ has integer knots $-k$, ..., k. By (6),

$$\begin{aligned} \phi _{2k+1} (t) = w_{t}^{2k+1,0} \phi _{2k} (t+0.5) + w_{t}^{2k+1,1} \, \phi _{2k} (t-0.5) \end{aligned}$$

and

$$\begin{aligned}&\phi _{2k} (t+0.5)= w_{t+0.5}^{2k,0} \, \phi _{2k-1} (t+1) + w_{t+0.5}^{2k,1} \, \phi _{2k-1} (t), \\&\phi _{2k} (t-0.5)= w_{t-0.5}^{2k,0} \, \phi _{2k-1} (t ) + w_{t-0.5}^{2k,1} \, \phi _{2k-1} (t-1). \end{aligned}$$

It means that $\phi _{2k+1}(t)$ can be expressed as

$$\begin{aligned} \omega _1 \, \phi _{2k-1}(t+1) + \omega _0 \, \phi _{2k-1} (t) + \omega _{-1}\, \phi _{2k-1}(t-1), \end{aligned}$$

where

$$\begin{aligned}&\omega _1 = w_{t}^{2k+1,0} w_{t+0.5}^{2k,0} ,\\&\omega _0 = w_{t}^{2k+1,0} w_{t+0.5}^{2k,1} + w_{t}^{2k+1,1} w_{t-0.5}^{2k,0},\\&\omega _{-1} = w_{t}^{2k+1,1} w_{t-0.5}^{2k,1} . \end{aligned}$$

After simplifying these expressions we obtain that

$$\begin{aligned} \phi _{2k+1}(t) = \sum _{i=-1}^{1} \omega _{i} \phi _{2k-1} (t + i), \end{aligned}$$

(8)

where

$$\begin{aligned} \omega _{\pm 1} = \frac{(k+1 \pm t)^2}{2k(2k+1)}, \quad \omega _0 = \frac{k^2 + k - t^2}{k(2k+1)}. \end{aligned}$$

(9)

Relations (8)–(9) together with (7) provide a way to construct recursively the central B-splines of odd degree.

Next we describe a construction of a generalized fuzzy partition, which will be based on the B-splines $\phi _{2k-1}$. Note that from (7) it follows that our construction generalizes the commonly used uniform fuzzy partition given by a generating function of a triangular shape (Perfilieva et al. 2011) (and, in case $k=1$, gives this particular uniform partition).

3.2 Construction of a generalized fuzzy partition based on the central B-spline of odd degree

Fix $N, k \in \mathbb {N}$ such that $N \ge 2k-1$. By A we denote the central B-spline of degree $2k-1$.

To construct fuzzy partition based on B-splines, we use the technique introduced in Stefanini (2011) and take basic functions $A_i$ which cover more than two consecutive subintervals (in our case, each basic function will cover 2k subintervals). This technique also requires choosing additional nodes $t_i$ with indices $i<0$ and $i>N$.

Let an interval [a, b] be fixed. Define

• the step: $h = (b-a)/(N+2k)$;

• the h-equidistant nodes: $t_i = a + h\, (i+k)$, for each $i \in [-k \, ..\, N+k]$; notice that

  $$\begin{aligned} a = t_{-k}< \cdots< t_0< \cdots< t_N< \cdots < t_{N+k} = b; \end{aligned}$$

• the basic functions $A_i(t) := A \left( \frac{t-t_i}{h} \right)$, $i\in [0 \, ..\, N]$.

We consider the (generalized) fuzzy partition $\mathbb {A}$ of [a, b] formed by the functions $A_{0}$, ..., $A_{N}$. Notice that this partition is (h, hk) -uniform.

From now on, we will call such partition of a given interval [a, b] as the (k, N) -FPB (fuzzy partition based on B-splines). When talking about such fuzzy partition, we also always implicitly assume that its parameters satisfy $N \ge 2k-1$.

3.3 Properties of the (k, N) -FPB

First we note that the support of each basic function $A_i$ consists of 2k consecutive basic intervals:

$$\begin{aligned} {{{\mathrm{supp}}}}(A_i) = \left( t_{i-k}, t_{i+k} \right) :=E_i, \quad i \in [0 \, ..\, N]. \end{aligned}$$

Denote

$$\begin{aligned} t_{k-1} = \hat{a}, \ t_{N-k+1} = \hat{b} , \end{aligned}$$

(10)

then $[\hat{a},\hat{b}] \subset [a,b]$ and for all $t \in [\hat{a},\hat{b}]$ we have

$$\begin{aligned} \sum _{i=0}^{N} A_i(t) = \sum _{i=0}^{N} A \left( x -i \right) = 1, \ \ x:= \frac{t-t_0}{h}, \end{aligned}$$

where the last equality follows from (5). It follows that for the constructed fuzzy partition the Ruspini condition is fulfilled on $[\hat{a}, \hat{b}]$. Notice that from the assumption $N \ge 2k-1$ it follows that $[\hat{a}, \hat{b}]$ always consists of at least one basic interval.

From the viewpoint of fuzzy partitions, the next lemma uses only the fact that the partition is uniform. It will help to understand the relations between different components of the direct $F^m$-transform when it is applied to a polynomial.

Lemma 3

Let $r\ge 0$ be an integer and $p^0$, $p^1$, ..., $p^r$ be polynomials satisfying $\deg p^l = l$ for each $l \in [0 \, ..\, r]$. Denote $p^l_i (t) := p^l \left( \frac{t - t_i}{h} \right)$, $i \in [0 \, ..\, N]$, $l \in [0 \, ..\, r]$.

Fix any $f \in \mathbb {P}_r$; let $a_{i,l}$, for all $i \in [0 \, ..\, N]$ and $l\in [0 \, ..\, r]$, be real numbers such that

$$\begin{aligned} f = \sum _{l=0}^{r} a_{i,l} p^l_i \end{aligned}$$

(such coefficients exist because $p^0 _i$, $p^1 _i$, ..., $p^r _i$ span the space $\mathbb {P}_r$).

Then for all $l \in [0 \, ..\, r]$ there exists a polynomial $q_l \in \mathbb {P}_{r-l}$ such that $a_{i,l} = q_l(i)$ for all $i \in [0 \, ..\, N]$.

Proof in Sect. 9.1.3.

Another lemma characterizes how orthogonality of polynomials w.r.t. the weight function A translates to a similar orthogonality-like property w.r.t. sum over the values of the corresponding expression. This property is crucial to our main results, since it easily leads to Lemma 7, which in turn is the main result behind Theorem 1.

Lemma 4

Suppose that $l \in \mathbb {N}$ and a polynomial $P \in \mathbb {P}_l$ satisfies

• 1. $\int _{-k}^{k} q(t) P(t)A(t)\,\mathrm {d}t=0$ for all polynomials q of degree at most $\min \left\{ l-1, 2k-1-l \right\}$;

• 2. P is either even or odd (i.e., for all $t \in \mathbb {R}$ we have $P(t) = \pm P(-t)$).

Let $p \in \mathbb {P}_{l-1}$ and suppose that $\deg p \le 2k-1-l$. Then for all $\tau \in [0,1]$ and $l \in [0 \, ..\, 2k-1]$ the equality

$$\begin{aligned} \sum _{j=1-k}^{k} p(j) P(\tau -j) A(\tau -j) = 0 \end{aligned}$$

holds.

Proof in Sect. 9.3.2.

3.4 Examples of FPBs

In Fig. 1 the (1, 8) -FPB of the interval [0, 1] is depicted. Since $k=1$, the basic functions are degree-1 B-splines, i.e., piecewise linear functions, commonly called triangular basic functions.

In Fig. 2 we show the (2, 8) -FPB of [0, 1] . Here $k=2$, thus the basic functions are given by cubic B-splines and each basic functions cover four consecutive basic intervals.

In both cases $N=8$, meaning that the unit interval is partitioned into 9 fuzzy sets. We also mark the endpoints of the interval where the Ruspini condition is fulfilled:

• for the (1, 8) -FPB of [0, 1] (Fig. 1), the Ruspini condition holds on [1 / 10; 9 / 10] ;

• for the (2, 8) -FPB of [0, 1] (Fig. 2), the Ruspini condition holds on [1 / 4; 3 / 4] .

Fig. 1

The (1, 8) -FPB of the unit interval (the basic functions are triangular-shaped)

Fig. 2

The (2, 8) -FPB of the unit interval (the basic functions are cubic B-splines)

3.5 Discrete grid associated to the FPB

For the purposes of the discrete transform we will define the discrete grid where the discrete functions will be defined. Fix a positive integer M and let

• $L = (N+2k)M$;

• $h' = h/M$;

• $z_{j} = a + h'\,j$, for all $j \in [0 \, ..\, L]$.

Then

$$\begin{aligned} z_{0} = t_{-k} =a \quad \text {and}\quad z_{L} = t_{N+k} = b. \end{aligned}$$

Denote

Notice that $\varDelta \cup \left\{ a,b \right\}$ is a uniform grid which contains all fuzzy partition nodes $t_i$, $i\in [-k \, ..\, N+k]$.

In Fig. 3, we depict the (1, 4) -FPB of the interval [0; 1] with its basic nodes (the circles in the figure) and show the discrete set $\varDelta$ (the asterisks in the figure). In this case $M=2$, i.e., each basic interval is split into two equal parts by the discrete points. It can also be noted that each fuzzy set $A_i$ contains exactly $2kM-1 = 3$ discrete points with nonzero degree.

Fig. 3

Illustration of the basic nodes and discrete data points, $k=1$, $N = 4$ and $M=2$

4 Spaces related to the fuzzy transforms

4.1 Overview

As previously, we will denote by $L_2(\mathbb {A})$ the space of all functions $f : [a,b] \rightarrow \mathbb {R}$ such that the restriction of f onto the support of the ith basic function $A_{ i}$ belongs to $L_2^{}(A_{i})$ for all $i\in [0 \, ..\, N]$. In other words, $L_2(\mathbb {A})$ contains all functions for which the integral $F ^m$-transform w.r.t. the (k, N) -FPB can be applied.

Similarly, by $\varLambda (\mathbb {A})$ we will denote the space of all functions $f : \varDelta \rightarrow \mathbb {R}$, i.e., all those functions for which the discrete $F ^m$-transform w.r.t. the (k, N) -FPB can be applied (provided that $\varDelta$ is sufficiently dense in the FPB).

In either the integral or the discrete case function f, to which the direct transform is applied, must satisfy certain properties; in particular, the restriction must belong to a certain space (in the integral case that is the space $L_2^{}(A_{i})$).

In this section we will describe both for the integral and discrete transforms a “template” space (either $L_2^{}(A)$ or $\varLambda (A)$, respectively), having the property that (in the integral case) all functions in the space $L_2^{}(A_{i})$ can be obtained from those in the template space by shifting and scaling the variables.

More precisely, $f \in L_2^{}(A_{i})$ iff there is a function $F \in L_2^{}(A)$ such that we have $F(x) = f \left( h x + t_{i} \right)$ for all $x \in {{\mathrm{supp}}}A$. The bijection $x \mapsto h x + t_{i}$ allows to describe objects in the space $L_2^{}(A_{i})$ (such as the inner product, norm, orthogonal polynomials, etc) in terms of the corresponding objects in $L_2^{}(A)$.

In the discrete case we define spaces $\varLambda (A_{i})$, the discrete counterparts of $L_2^{}(A_{i})$, and a template space $\varLambda (A)$, from which the space $\varLambda (A_{i})$ can be derived (via the bijection $x \mapsto h x + t_{i}$).

4.2 Spaces related to the integral transform

4.2.1 Template space $L_2^{}(A)$

Let $L_2^{}(A)$ stand for the Hilbert space of square integrable functions $f : [-k,k] \rightarrow \mathbb {R}$ with the inner product $\left\langle \cdot ,\cdot \right\rangle$ given by

$$\begin{aligned} \left\langle f,g \right\rangle = \int \limits _{-k}^{k} f(t)g(t) A(t) \,\mathrm {d}t\end{aligned}$$

(11)

and the induced norm $\left\| f \right\| = \left\langle f,f \right\rangle ^{0.5}$.

There exists a sequence of monic orthogonal (w.r.t the inner product $\left\langle \cdot ,\cdot \right\rangle$) polynomials in $L_2^{}(A)$. This follows from a construction similar as in Perfilieva et al. (2011), or, more generally, from Gautschi (2004, Theorem 1.7). Moreover, these polynomials can be constructed via a three-term recurrence relation Gautschi (2004, Theorem 1.27) as follows:

$$\begin{aligned}&\mathtt {P}_{-1} \equiv 0, \quad \mathtt {P}_{0} \equiv 1; \\&\mathtt {P}_{l}(t) = (t- \alpha _l) \mathtt {P}_{l-1}(t) - \beta _{l-1} \mathtt {P}_{l-2}(t), \quad l \in \mathbb {N}, \end{aligned}$$

where

$$\begin{aligned} \alpha _l = \frac{ \left\langle t\mathtt {P}_{l-1},\mathtt {P}_{l-1} \right\rangle }{\left\| \mathtt {P}_{l-1} \right\| ^2}, \ \beta _{l} = \frac{ \left\| \mathtt {P}_{l} \right\| ^2}{\left\| \mathtt {P}_{l-1} \right\| ^2}, \quad l \in \mathbb {N}, \end{aligned}$$

and $\beta _0 \in \mathbb {R}$ is arbitrary. However, it is easy to see that $\alpha _l=0$; hence, the three-term recurrence relation in our case is

$$\begin{aligned} \mathtt {P}_{l}(t) =t \cdot \mathtt {P}_{l-1}(t) - \frac{ \left\| \mathtt {P}_{l-1} \right\| ^2}{\left\| \mathtt {P}_{l-2} \right\| ^2} \cdot \mathtt {P}_{l-2}(t), \quad l=2,3,\ldots , \end{aligned}$$

with initial members $\mathtt {P}_{0} \equiv 1$, $\mathtt {P}_{1}(t) =t$.

By inductive arguments it can be seen that $\mathtt {P}_{l}$ is an odd function iff l is odd and $\mathtt {P}_{l}$ is an even function iff l is even. Another obvious but important property is that $\mathtt {P}_{l}$ is orthogonal to every polynomial of degree less than l.

4.2.2 Orthogonal polynomials for $L_2^{}(A_{i})$

Let $f,g \in L_2^{}(A)$ and denote $f_i(x) = f\left( \frac{x-t_i}{h} \right)$ and $g_i(x) =g\left( \frac{x-t_i}{h} \right)$, then $f_i, g_i \in L_2^{}(A_{i})$ (conversely, if $\tilde{f}_i$ is an arbitrary function from $L_2^{}(A_{i})$, then $\tilde{f} (x) :=\tilde{f}_i \left( xh+t_i \right)$ is in $L_2^{}(A)$). The inner product $\left\langle \cdot ,\cdot \right\rangle _{i}$ can be obtained from $\left\langle \cdot ,\cdot \right\rangle$ as follows:

$$\begin{aligned} \left\langle f_i,g_i \right\rangle _{i}= & {} \int \limits _{t_{i-k}}^{t_{i+k}} f_i(t) g_i(t) A_i(t) \,\mathrm {d}t\\= & {} h \, \int \limits _{-k}^{k} f \left( x \right) g\left( x \right) A(x) \,\mathrm {d}x= h \, \left\langle f, g \right\rangle . \end{aligned}$$

It immediately follows that also $\left\| f \right\| =h^{-0.5} \left\| f_i \right\| _{i}$.

Then the basis of $L_2^m(A_i)$ is formed by the polynomials $\mathtt {P}_{i,l}$, $l\in [0 \, ..\, m]$, where

$$\begin{aligned} \mathtt {P}_{i,l}(t) = \mathtt {P}_{l} \left( \frac{t -t_i}{h} \right) . \end{aligned}$$

(12)

Moreover, $\left\| \mathtt {P}_{i,l} \right\| ^2_i = h \,\left\| \mathtt {P}_{l} \right\| ^2$.

Finally, for all $f \in L_2^{}(A_{i})$, $i\in [0 \, ..\, N]$, and all $l \ge 0$ consider $c_{i,l}[f]$, defined by (2). This quantity has clear interpretation in terms of projection onto the basis polynomial $\mathtt {P}_{i,l}$: the orthogonal projection of f onto the 1-dimensional space spanned by $\mathtt {P}_{i,l}$ is the function $c_{i,l}[f] \mathtt {P}_{i,l}$.

4.3 Spaces related to the discrete transform

4.3.1 Template $Z$ and the associated space $\varLambda (A)$

For all integers j s.t. $\left| j \right| \le Mk -1$, introduce numbers . The set $Z$ has the following property:

In other words, by scaling points in $Z$ by h and shifting by $t_i$, we obtain all points in $\varDelta _i$, i.e., the set $Z$ can be thought of as a template for the discrete points in $\varDelta _i$.

Let $\varLambda (A)$ stand for the space of functions $f : Z\rightarrow \mathbb {R}$. Define a bilinear form $\left\langle \cdot ,\cdot \right\rangle : \varLambda (A)^2 \rightarrow \mathbb {R}$ with

$$\begin{aligned} \left\langle f,g \right\rangle = \sum _{\tilde{z} \in Z} f(\tilde{z})g(\tilde{z}) A(\tilde{z}) , \quad f,g\in \varLambda (A), \end{aligned}$$

(13)

and let $\left\| f \right\| = \left\langle f,f \right\rangle ^{0.5}$, for all $f \in \varLambda (A)$.

Identifying functions $f,g \in \varLambda (A)$ which are almost everywhere equal (in the sense $\left\| f-g \right\| = 0$) turns the bilinear form $\left\langle \cdot ,\cdot \right\rangle$ into an inner product and $\left\| \cdot \right\|$ into a norm.

Since $Z$ consists of $2Mk-1$ points (and at any of them $A$ takes a positive value), by Gautschi (2004, Theorem 1.7) there exist polynomials $\mathtt {P}_{l}$, $l \in [0 \, ..\, 2Mk-2]$, such that

• $\deg \mathtt {P}_{l} =l$;

• the polynomials are orthogonal with respect to $\left\langle \cdot ,\cdot \right\rangle$, i.e., $\left\langle \mathtt {P}_{l} ,\mathtt {P}_{l'} \right\rangle = 0$ whenever $l \ne l'$;

• $\left\| \mathtt {P}_{l} \right\| \ne 0$ for each $l \in [0 \, ..\, 2Mk-2]$;

• $\mathtt {P}_{l}$ is an even function if l is even and odd if l is odd.

Moreover, these polynomials can also be constructed via a three-term recurrence relation, see, e.g., Gautschi (2004, Theorem 1.27).

We extend this construction by defining $\mathtt {P}_{l} \in \mathbb {P}_l$ with $l \ge 2Mk-1$ so that $\mathtt {P}_{l} \equiv 0$ on $Z$, then the obtained infinite sequence is orthogonal w.r.t. $\left\langle \cdot ,\cdot \right\rangle$ (even though $\left\| \mathtt {P}_{l} \right\| =0$ for all $l \ge 2Mk-1$, thus these additional polynomials are identified with 0 in the space $\varLambda (A)$).

For each $l \ge 1$ the system $\left\{ \mathtt {P}_{0}, \mathtt {P}_{1}, \ldots ,\mathtt {P}_{l} \right\}$ spans the space $\mathbb {P}_l$; moreover, $\mathtt {P}_{l}$ is orthogonal (w.r.t. $\left\langle \cdot ,\cdot \right\rangle$) to any $p \in \mathbb {P}_{l-1}$.

4.3.2 Space $\varLambda (A_{i})$

Let $\varLambda (A_{i})$, $i \in [0 \, ..\, N]$, stand for the space of functions $f : \varDelta _i \rightarrow \mathbb {R}$. Clearly, if $f \in \varLambda (A)$, then $f_i$, defined by $f_i(x) = f \left( \frac{x-t_i}{h} \right)$, belongs to $\varLambda (A_{i})$ (and vice versa).

Each space $\varLambda (A_{i})$ is equipped with the discrete inner product defined by

$$\begin{aligned} \left\langle f,g \right\rangle _{i} = \sum _{t\in \varDelta _i } f(t)g(t) A_i(t) , \quad f,g \in \varLambda (A_{i}), \end{aligned}$$

(14)

and the associated discrete norm $\left\| \cdot \right\| _{i}$.

We define

$$\begin{aligned} \mathtt {P}_{i,l} (t) = \mathtt {P}_{l} \left( \frac{t - t_i}{h} \right) \end{aligned}$$

(15)

obtaining the sequence $\mathtt {P}_{i,l}$, $l \ge 0$, of polynomials orthogonal w.r.t. $\left\langle \cdot ,\cdot \right\rangle _{i}$.

Similarly as in the integral case (see Sect. 4.2.2), the inner product in the space $\varLambda (A_{i})$ is related to that in the space $\varLambda (A)$:

Lemma 5

For all $i \in [0 \, ..\, N]$ and $f,g \in \varLambda (A)$ the equality $\left\langle f_i,g_i \right\rangle _{i} = \left\langle f, g \right\rangle$ holds, where $f_i(x)$ and $g_i(x)$ stand for the functions $f \left( \frac{x-t_i}{h} \right)$ and $g\left( \frac{x-t_i}{h} \right)$, respectively.

Proof in Sect. 9.4.2.

It immediately follows that also $\left\| f \right\| = \left\| f_i \right\| _{i}$.

Finally, for all $f \in \varLambda (A_{i})$ , $i\in [0 \, ..\, N]$, and all $l \in [0 \, ..\, 2kM-2]$ we define

$$\begin{aligned} c_{i,l}[f] = \frac{\left\langle f,\mathtt {P}_{i,l} \right\rangle _{i}}{\left\| \mathtt {P}_{i,l} \right\| _{i}^2}. \end{aligned}$$

(16)

If no confusion can appear, this notation is simplified to $c_{i,l}$. Again, the orthogonal projection of f onto the 1-dimensional space spanned by $\mathtt {P}_{i,l}$ is $c_{i,l}[f] \mathtt {P}_{i,l}$.

4.4 Relationship and common properties

From now on we will denote by

• ${F}_m^\rightarrow$ either the integral or the discrete direct $F^m$-transform (and $F_{m,i}^\rightarrow$ stands for the ith component of the direct transform in either case);

• ${F}_m^\leftarrow$ either the integral or the discrete inverse $F^m$-transform;

• $\left\langle \cdot ,\cdot \right\rangle$ either the inner product in the space $L_2^{}(A)$, defined by (11), or the inner product in the space $\varLambda (A)$, defined by (13);

• $\left\langle \cdot ,\cdot \right\rangle _{i}$ either the inner product in the space $L_2^{}(A_{i})$, defined by (1), or the inner product in the space $\varLambda (A_{i})$, defined by (14);

• $\left\| \cdot \right\|$ the norm, associated to $\left\langle \cdot ,\cdot \right\rangle$ (in either the discrete or the integral case); likewise, $\left\| \cdot \right\| _{i}$ stands for the norm, associated to $\left\langle \cdot ,\cdot \right\rangle _{i}$;

• $\mathtt {P}_{l}$ the orthogonal polynomials in either $L_2^{}(A)$ or $\varLambda (A)$;

• $\mathtt {P}_{i,l}$ the orthogonal polynomials in either $L_2^{}(A_{i})$ or $\varLambda (A_{i})$;

• $c_{i,l}[f]$ (or $c_{i,l}$) the scalars, defined either by (2) or by (16).

Whether the integral or the discrete construction is meant will be clear from the context; in certain cases (such as in Theorem 1) notations ${F}_m^\rightarrow$, $\mathtt {P}_{l}$, etc., can stand for either the integral or the discrete concepts, allowing to unify the results obtained in both settings.

However, when confusion can appear (as in Lemma 6 below, where both inner products appear) the integral and discrete concepts will be distinguished by use of superscripts ${}^{\mathrm {int}}$ and ${}^{\mathrm {dsc}}$, respectively; for example, $\mathtt {P}_{l}^{\mathrm {int}}$ stands for the polynomial $\mathtt {P}_{l}$ defined in Sect. 4.2, whereas $\mathtt {P}_{l}^{\mathrm {dsc}}$ stands for the polynomial $\mathtt {P}_{l}$ described in Sect. 4.3.

The next lemma shows that the weighted integral (w.r.t. the weight function A) of a polynomial of degree at most $2k-1$ can be replaced with a sum over the points in the template $Z$, which establishes a relationship between the integral and discrete inner products defined previously:

Lemma 6

For all $p \in \mathbb {P}_{2k-1}$ the following equality holds:

$$\begin{aligned} \int \limits _{\mathbb {R}} p(t) A(t) \,\mathrm {d}t= \frac{1}{M}\sum _{j=-Mk+1}^{Mk-1} p(\tilde{z}_{j}) A(\tilde{z}_{j}) . \end{aligned}$$

In terms of the inner products $\left\langle \cdot ,\cdot \right\rangle ^{\mathrm {int}}$ and $\left\langle \cdot ,\cdot \right\rangle ^{\mathrm {dsc}}$ this can be restated as

$$\begin{aligned} \left\langle p,q \right\rangle ^{\mathrm {int}}= \frac{1}{M} \left\langle p,q \right\rangle ^{\mathrm {dsc}}, \text { for all } p,q \in \mathbb {P} \text { s.t. } pq \in \mathbb {P}_{2k-1} . \end{aligned}$$

Proof in Sect. 9.4.1.

Finally, we have the following result, which is a consequence of Lemma 4; as mentioned previously, this is the main result leading to Theorem 1.

Lemma 7

Let $l \in [1 \, ..\, 2k-1]$. Suppose that p is any polynomial such that $\deg p \le \min \left\{ l-1, 2k-1-l \right\}$. Then for all $\tau \in [0,1]$ the following equality holds:

$$\begin{aligned} \sum _{j=1-k}^{k} p(j) \cdot \mathtt {P}_{l}(\tau -j) A(\tau -j) = 0. \end{aligned}$$

Proof

Fix $l \in [1 \, ..\, 2k-1]$. We claim that $\mathtt {P}_{l} \in \mathbb {P}_l$ is orthogonal under the integral inner product $\left\langle \cdot ,\cdot \right\rangle ^{\mathrm {int}}$ to all polynomials q of degree at most $\min \left\{ l-1, 2k-1-l \right\}$. Then, since $\mathtt {P}_{l} \in \mathbb {P}_l$ is either an odd or an even function, the claim follows from Lemma 4.

In the integral case we already have the required orthogonality; consider the discrete case and fix any polynomial $q \in \mathbb {P}_{l-1}$ s.t. $\deg q \le 2k-1-l$. Then $\deg (q \mathtt {P}_{l}) \le 2k-1$ and, by Lemma 6,

$$\begin{aligned} \left\langle q, \mathtt {P}_{l} \right\rangle ^{\mathrm {int}}= \frac{1}{M} \left\langle q, \mathtt {P}_{l} \right\rangle ^{\mathrm {dsc}}. \end{aligned}$$

However, since q is orthogonal to $\mathtt {P}_{l}$ under $\left\langle \cdot ,\cdot \right\rangle ^{\mathrm {dsc}}$, it follows that $\int _{-k}^k q(x) \mathtt {P}_{l}(x) A(x)\,\mathrm {d}x=0$ for all polynomials q of degree at most $\min \left\{ l-1, 2k-1-l \right\}$, concluding the proof. $\square$

5 Properties of the discrete $F^m$-transform w.r.t. FPB

Throughout this section, fix a function $f : \varDelta \rightarrow \mathbb {R}$; here the set $\varDelta$ is as described in Sect. 3.5. Denote

$$\begin{aligned} \mathbf {Y} = \begin{pmatrix} f(z_{1})\\ \vdots \\ f(z_{L-1}) \end{pmatrix}, \qquad \mathbf {Y}_i = \begin{pmatrix} f(h\tilde{z}_{-Mk+1}+t_i)\\ \vdots \\ f(h\tilde{z}_{Mk-1}+t_i) \end{pmatrix}, \end{aligned}$$

for all , the vector $\mathbf {Y}_i$ is the fragment of $\mathbf {Y}$ which contains the discrete function’s values on $\varDelta _i$.

The following lemma helps to characterize the values of m which ensures sufficient density of $\varDelta$.

Lemma 8

Let $m \ge 0$. Then $\varDelta$ is sufficiently dense in the (k, N) -FPB w.r.t. a nonnegative integer m iff $m \le 2kM -2$.

Proof in Sect. 9.5.2.

It also can be shown that the coefficients of the polynomial $F_{m,i}^\rightarrow [f]$ can be computed via matrices, independent of i (except for the corresponding fragment of the vector $\mathbf {Y}$), which is due to the uniformness; moreover, the Vandermonde matrix $\mathbf {X}$ can be formed by any polynomials spanning $\mathbb {P}_m$.

Lemma 9

Let $m \in [0 \, ..\, 2Mk-2]$ and $p^0$, $p^1$, ..., $p^m$ be arbitrary polynomials spanning $\mathbb {P}_m$.

Denote $p^l_i (t) := p^l \left( \frac{t - t_i}{h} \right)$ and define matrices $\mathbf {X}$, $\mathbf {A}$ as follows:

$$\begin{aligned}&\mathbf {X} = \begin{pmatrix} p^l(\tilde{z}_{r}) \end{pmatrix}_{r,l}, \ r \in [-Mk+1,Mk-1], \ l \in [0 \, ..\, m]; \\&\mathbf {A} = {{\mathrm{diag}}}\left( A(\tilde{z}_{-Mk+1}), \ldots , A(\tilde{z}_{Mk-1}) \right) . \end{aligned}$$

Then the components of the discrete direct transform ${F}_m^\rightarrow [f]$ satisfy

$$\begin{aligned} F_{m,i}^\rightarrow {}[f](t) = \sum _{l=0}^{m} \beta ^{(i)}_l p^l_i(t), \end{aligned}$$

where

$$\begin{aligned} \beta ^{(i)}= \left( \mathbf {X} ^T \mathbf {A} \mathbf {X} \right) ^{-1} \mathbf {X} ^T \mathbf {A} \mathbf {Y}_i . \end{aligned}$$

Proof in Sect. 9.5.3.

Alternatively, the components of the discrete $F^m$-transform of f can be computed similarly as in the case of the integral transform.

Corollary 1

Suppose that $m \in [0 \, ..\, 2Mk-2]$. Then the components of the discrete direct transform ${F}_m^\rightarrow [f]$ satisfy

$$\begin{aligned} F_{m,i}^\rightarrow {}[f](t) = \sum _{l=0}^m c_{i,l} \mathtt {P}_{i,l} \left( t \right) , \quad i \in [0 \, ..\, N], \end{aligned}$$

(17)

where $c_{i,l}$ are defined by (16).

Proof

We apply Lemma 9 with $p^l = \mathtt {P}_{l}$. As in the proof of Lemma 8, notice that $\mathbf {X}^T \mathbf {A} \mathbf {X}$ is the diagonal matrix with entries $\left\| \mathtt {P}_{l} \right\| ^2$, $l \in [0 \, ..\, m]$, on its diagonal. Similarly, the lth entry of $\mathbf {X}^T \mathbf {A} \mathbf {Y}_i$ equals

$$\begin{aligned}&\sum _{\tilde{z} \in Z} \mathtt {P}_{l} (\tilde{z}) A(\tilde{z}) f \left( h\tilde{z} + t_i \right) =\\&\quad = \sum _{z \in \varDelta _i} \mathtt {P}_{i,l} (z) A_i(z) f(z) = \left\langle \mathtt {P}_{i,l} ,f \right\rangle _{i}. \end{aligned}$$

Thus the lth entry of $(\mathbf {X}^T \mathbf {A} \mathbf {X} ) ^{-1} \mathbf {X}^T \mathbf {A} \mathbf {Y}_i$ is

$$\begin{aligned} \frac{ \left\langle f,\mathtt {P}_{i,l} \right\rangle _{i} }{\left\| \mathtt {P}_{l} \right\| ^2 } = \frac{ \left\langle f,\mathtt {P}_{i,l} \right\rangle _{i} }{\left\| \mathtt {P}_{i,l} \right\| _{i}^2 } = c_{i,l} , \end{aligned}$$

concluding the proof. $\square$

We also obtain the following conclusion, the discrete counterpart of Lemma 1.

Corollary 2

Suppose that $f \in \mathbb {P}_{r}$, for some $r \in \mathbb {N}_0$, is expressed as

$$\begin{aligned} f = \sum _{l=0}^{r} \alpha _{l} \mathtt {P}_{i,l} \end{aligned}$$

for some $i \in [0 \, ..\, N]$ and real coefficients $\alpha _{l}$. Denote $\alpha _l = 0$ for all $l > r$.

Let $m \le 2kM-2$; then $c_{i,l}[f] = \alpha _l$ for all $l \in [0 \, ..\, m]$ and, consequently, the ith component of the discrete direct transform $F^m$-transform equals

$$\begin{aligned} F_{m,i}^\rightarrow [f](t ) = \sum _{l=0}^m \alpha _{l} \mathtt {P}_{i,l} (t). \end{aligned}$$

Proof

Apply Corollary 1 and from the orthogonality of $\mathtt {P}_{i,l}$, $l \in [0 \, ..\, m ]$, obtain that

$$\begin{aligned} \left\langle f,\mathtt {P}_{i,l} \right\rangle _{i} = \alpha _{l}\left\| \mathtt {P}_{i,l} \right\| _{i}^2 . \end{aligned}$$

Since $\left\| \mathtt {P}_{i,l} \right\| _{i} = \left\| \mathtt {P}_{l} \right\| \ne 0$ when $l \le m \le 2kM-2$, it follows that $c_{i,l}[f] = \alpha _l$. $\square$

6 Additional properties of $F^m$-transforms

In this section we show several additional results related to $F^m$-transforms, when the original function f is of a particular kind. Even though we formulate and prove the following results in the case of FPB, the choice of the fuzzy partition is not essential and similar results could be obtained in a more general setting.

Throughout this section, let numbers $k, N \in \mathbb {N}$, an interval [a, b] and its (k, N) -FPB, given by the basic functions $A_i(x) = A\left( \frac{x-t_i}{h} \right)$, be fixed. Suppose that the Ruspini condition is fulfilled on an interval $[\hat{a}, \hat{b}] \subset [a,b]$.

Fix also a function $f : D_f \rightarrow \mathbb {R}$, where $D_f \subset [a,b]$ is

• either the whole interval [a, b] (in the case of the integral $F^m$-transform; then we also require f to be from the space $L_2(\mathbb {A})$),

• or the discrete grid $\varDelta$, described in Section 3.5 (in the case of the discrete $F^m$-transform; then we also require $m \le 2kM-2$).

6.1 $F^m$-transform of an integral transform

We consider the case when f is itself an integral transform of a function g w.r.t. some kernel K (in our consideration kernel K is piecewise continuous w.r.t. each variable). Then the $F^m$-transform of f can be obtained, instead applying the $F^m$-transform to the function g.

Lemma 10

Suppose that

$$\begin{aligned} f(x) = \int \limits _{a}^{b} g(t) \, K(x,t) \,\mathrm {d}t, \quad x \in D_f, \end{aligned}$$

where $g \in L_1([a,b] )$ and $K: [a,b] ^2 \rightarrow \mathbb {R}$ is piecewise continuous w.r.t. each variable. Then

$$\begin{aligned} {F}_m[f](x) = \int \limits _{a}^{b} g(t) \, {F}_m[K( \cdot ,t)] (x) \,\mathrm {d}t, \quad x \in [\hat{a}, \hat{b}] . \end{aligned}$$

Proof in Sect. 9.6.1.

6.2 $F^m$-transform of a hybrid function

In this subsection we consider the case when the original function is defined in a piecewise manner and show that its $F^m$-transform can be obtained as the $F^m$-transform of the corresponding sub-functions, as long as the argument is far enough from the point between pieces.

Lemma 11

Suppose that $s \in [t_{i_0}, t_{i_0+1}]$ for some $i_0 \in [-k \, ..\, N+k-1]$. Let f satisfy

$$\begin{aligned} f(x)= {\left\{ \begin{array}{ll} g_1(x), &{} x \le s, \\ g_2(x), &{} x > s, \end{array}\right. } \end{aligned}$$

for some functions $g_1, g_2 : D_f \rightarrow \mathbb {R}$.

Let $x \in [t_{j_0}, t_{j_0+1}] \subset [\hat{a}, \hat{b}]$ for some integer $j_0$. Then the $F^m$-transform of f satisfies the following properties:

• if $j_0 \le i_0 - 2k$, then ${F}_m[f] (x) = {F}_m[g_1](x)$;

• if $j_0 \ge i_0 + 2k$, then ${F}_m[f] (x) = {F}_m[g_2](x)$.

Proof in Sect. 9.6.2.

6.3 Affine transformations of the universe

In this subsection we consider the case when the original interval is transformed to another interval via an affine map $\phi (t) = \alpha t + \beta$, where $\alpha , \beta$ are some reals, $\alpha > 0$.

Clearly, we immediately have the (k, N) -FPB of the interval $[ \phi (a), \phi (b)]$, with basic nodes $\phi (t_i)$ and basic functions

$$\begin{aligned} \tilde{A}_i (u) = A\left( \frac{u-\phi (t_i)}{\phi (h)} \right) , \quad u \in [ \phi (a), \phi (b)] , \end{aligned}$$

for all $i \in [-k \, ..\, N+k]$. Notice that alternatively we can write

$$\begin{aligned} \tilde{A}_i (u) = A\left( \frac{\phi ^{-1}(u)-t_i}{h} \right) = A_i( \phi ^{-1}(u)), \end{aligned}$$

i.e., $\tilde{A}_i = A_i \circ \phi ^{-1}$. Clearly, this FPB fulfills the Ruspini condition on $[\phi (\hat{a}), \phi (\hat{b})]$.

In the case of the discrete transform, the set $\varDelta \subset [a,b]$ is transformed to the set $\tilde{\varDelta } = \phi (\varDelta )$. If $\varDelta$ is a uniform grid as described in Sect. 3.5 (with parameter M), then $\tilde{\varDelta }$ is also a uniform grid with the same parameter M.

It can be seen that the basis polynomials $\mathtt {P}_{i,l}(x)$, corresponding to the FPB of [a, b] , are transformed to polynomials $\tilde{\mathtt {P}}_{i,l}(u)$ for the respective FPB of $[\phi (a), \phi (b)]$ via

$$\begin{aligned} \tilde{\mathtt {P}}_{i,l} = \mathtt {P}_{i,l} \circ \phi ^{-1}. \end{aligned}$$

For a function g (either in $L_2(\tilde{A}_i)$ or defined over $\tilde{\varDelta }_i$), the scalar, defined similarly as in (2) (or (16)) w.r.t. the (k, N) -FPB of $[\phi (a),\phi (b)]$, will be denoted by $\tilde{c}_{i,l}[g]$.

Finally, we denote the $F^m$-transform of a function g w.r.t. the (k, N) -FPB of $[\phi (a), \phi (b)]$ by $F_{m,\phi } [g]$. Then we claim that

Lemma 12

Let $g = f \circ \phi ^{-1}$. Then the following equality holds on $[\phi (\hat{a}), \phi (\hat{b})]$:

$$\begin{aligned} F_{m,\phi } [g]= {F}_m[f] \circ \phi ^{-1} . \end{aligned}$$

Proof in Sect. 9.6.3.

As a consequence, we have the following result.

Corollary 3

Let $\alpha >0$ and $\beta$ be scalars and denote $\phi (t) = \frac{t - \beta }{\alpha }$. Suppose there is another scalar $\gamma _\alpha$ (which may depend on $\alpha$) and a function g (defined on $\phi (D_f)$) such that f and g satisfy

$$\begin{aligned} f(\alpha u + \beta ) = \gamma _\alpha g(u) , \quad u \in \phi (D_f). \end{aligned}$$

Then

$$\begin{aligned} {F}_m[f](x) = \gamma _\alpha F_{m,\phi } [g] \left( u \right) , \quad x \in [a,b], \ u:= \phi (x) , \end{aligned}$$

where $F_{m,\phi } [g]$ stands for the $F^m$-transform of g w.r.t. the (k, N) -FPB of the interval $\left[ \phi (a) , \phi (b) \right]$.

Proof

Apply the preceding lemma to obtain

$$\begin{aligned} {F}_m[f](x) = F_{m,\phi } [\tilde{f}](u), \quad x\in [a,b], \end{aligned}$$

where $\tilde{f} := f \circ \phi ^{-1}$ and $u := \phi (x)$. Notice that

$$\begin{aligned} \tilde{f}(u) = f \left( \phi ^{-1}(u) \right) = f(x) = \gamma _\alpha g(u), \end{aligned}$$

i.e., $\tilde{f} = \gamma _\alpha \cdot g$. Now from the linearity of the $F^m$-transforms we conclude $F_{m,\phi } [\tilde{f}] = \gamma _\alpha F_{m,\phi } [g]$. $\square$

We are interested to apply this corollary to the case when [a, b] consists of only $4k-1$ basic intervals and the affine transformation maps it to the symmetric interval $[-2k + 0.5, 2k-0.5]$. For these purposes, we denote

• by $\mathcal {I}_k$ the interval $[-2k + 0.5, 2k-0.5]$;

• by $F_{m,*}$ the $F^m$-transform w.r.t. the $(k,2k-1)$-FPB of the interval $\mathcal {I}_k$.

Notice that the FPB of $\mathcal {I}_k$ has unit step (i.e., independent of the initial parameter h, characterizing the FPB of [a, b] ).

7 Approximation of polynomials using $F^m$-transform based on B-splines

Throughout the next two sections, we fix an interval $[a,b] \subset \mathbb {R}$ and its (k, N) -FPB (for some $k,N \in \mathbb {N}$), denoted by $\mathbb {A}$. For the discrete transform we suppose that also a uniform discrete grid $\varDelta$ is fixed as described in Sect. 3.5 (with the corresponding parameter M).

Theorem 1

Let r, m be nonnegative integers satisfying $r \le \min \left\{ 2m+1, 2k-1 \right\}$. Suppose that $f \in \mathbb {P}_{r}$; then

$$\begin{aligned} f(t) = {F}_m[f](t) \quad \text {for all}\quad t\in [\hat{a},\hat{b}]. \end{aligned}$$

Here $[\hat{a},\hat{b}]$ is defined by (10) and ${F}_m[f]$ stands for either the integral or the discrete $F^m$-transform of f w.r.t. $\mathbb {A}$. In the discrete case also $m \le 2kM-2$ is required so that $\varDelta$ is sufficiently dense.

Proof

Fix an arbitrary $f \in \mathbb {P}_{r}$. Assume that $r > m$, for otherwise the claim trivially follows from the $F^m$-transform properties.

Since polynomials ${\mathtt {P}_{i,0}, \ldots , \mathtt {P}_{i,r}}$ for any $i \in [0 \, ..\, N]$ form a basis of $\mathbb {P}_r$, the polynomial f can be expressed as

$$\begin{aligned} f (t) = \sum _{l=0}^{r} c_{i,l} \mathtt {P}_{i,l}(t) \end{aligned}$$

for some real coefficients $c_{i,l}$, $l \in [0 \, ..\, r]$, $i \in [0 \, ..\, N]$.

Now, by Lemma 1 (or Corollary 2, respectively), $c_{i,l}$ are defined by (2) (or (16), respectively), and the ith component of ${F}_m^\rightarrow [f]$ equals

$$\begin{aligned} F_{m,i}^\rightarrow {}[f](t) = \sum _{l=0}^m c_{i,l} \mathtt {P}_{i,l}(t), \quad i \in [0 \, ..\, N] . \end{aligned}$$

Thus, the $F^m$-transform of f is given by

$$\begin{aligned} {F}_m{}[f](t) = \sum _{i=0}^{N } \sum _{l=0}^m c_{i,l} \mathtt {P}_{i,l}(t) A_i(t) . \end{aligned}$$

To prove the theorem, it is sufficient to show that for all ${ t \in [\hat{a},\hat{b}]}$ and ${l \in [m+1 \, ..\, r] }$ the equality

$$\begin{aligned} \sum _{i=0}^{N} c_{i,l} \mathtt {P}_{i,l}(t) A_i(t) = 0 \end{aligned}$$

(18)

holds.

From (12) and (15) the following equality is obtained:

$$\begin{aligned} \sum _{l=0}^{r} c_{i,l} \mathtt {P}_{i,l}(t) = \sum _{l=0}^{r} c_{i,l} \mathtt {P}_{l} \left( \frac{t -t_i}{h} \right) \end{aligned}$$

for all $i \in [0 \, ..\, N]$ and $t \in \mathbb {R}$.

Fix any $l \in [m+1 \, ..\, r]$. By applying Lemma 3 with $p^l = \mathtt {P}_{l}$, we obtain that there is a polynomial $q_l \in \mathbb {P}_{r-l}$ s.t. ${ c_{i,l} = q_l(i) }$ for all $i \in [0 \, ..\, N]$.

Notice that $\deg q_l \le r-l \le 2k-1-l$ satisfies also

$$\begin{aligned} \deg q_l \le r- (m+1) \le m \le l-1. \end{aligned}$$

Next fix any $t \in [\hat{a}, \hat{b}]$. There exists such an index $j \in [k-1 \, ..\, N-k]$ that $t \in [t_{j}, t_{j+1}]$ (remark: it is possible that there are several such indices j, i.e., $t=t_i$ for some $i \in [k \, ..\, N-k-1]$; then any of these indices can be chosen, say, $j=i$).

Then $A_i(t) = 0$ unless $i \in [j-k+1 \, ..\, j+k]$, thus

$$\begin{aligned} \sum _{i=0}^{N} c_{i,l} \mathtt {P}_{i,l}(t) A_i(t) = \sum _{i=j+1-k}^{j+k} c_{i,l} \mathtt {P}_{i,l}(t) A_i(t). \end{aligned}$$

The right side of this equality, by the definition of $A_i$ and either (12) or (15), can be rewritten as

$$\begin{aligned} \sum _{i=j+1-k}^{j+k} q_{l}(i) \, \mathtt {P}_{l} \left( \frac{t -t_i}{h} \right) A \left( \frac{t -t_i}{h} \right) . \end{aligned}$$

Denote $\tau = (t - t_j)/h \in [0,1]$, then $\left( t-t_{i+j} \right) /{h}= \tau -i$ for all $i \in [1-k \, ..\, k]$ and we have obtained that

$$\begin{aligned} \sum _{i=0}^{N} c_{i,l} \mathtt {P}_{i,l}(t) A_i(t) = \sum _{i=1-k}^{k} q_{l}(j+i) \, \mathtt {P}_{l} \left( \tau - i \right) A \left( \tau - i \right) . \end{aligned}$$

Let $p(x) = q_l(j+x)$, then its degree is $\deg p =\deg q_l$, thus $\deg p\le \min \left\{ l-1, 2k-1-l \right\}$ and, by Lemma 7,

$$\begin{aligned} \sum _{i=0}^{N} c_{i,l}\mathtt {P}_{i,l}(t) A_i(t) = \sum _{i=1-k}^{k} p(i) \, \mathtt {P}_{l} \left( \tau - i \right) A \left( \tau - i \right) =0, \end{aligned}$$

proving (18). Now it follows that for all $t \in [\hat{a},\hat{b}]$ the $F^m$-transform of f can indeed be expressed as

$$\begin{aligned} \sum _{j=0}^{N } \sum _{l=0}^{r} c_{j,l} \mathtt {P}_{j,l}(t) A_j(t) = \sum _{j=0}^{N } f(t) A_j(t) = f(t). \end{aligned}$$

Here the last equality follows from the Ruspini condition.

$\square$

8 Approximation properties of the $F^m$-transform based on FPB

8.1 Estimation of the approximation error

The next theorem gives an estimation of the quality of approximation of a function f from $L^{r+1}_q([a,b ])$ (i.e., a function s.t. $f^{(r+1)} \in L^q([a,b ])$, $q \ge 1$) by its $F^m$-transform ($r \le \min \left\{ 2m+1,2k-1 \right\}$) w.r.t. the (k, N) -FPB of [a, b] .

We shall denote ($n\in [0 \, ..\, r]$)

$$\begin{aligned} D_{n,m}[f](x) = \frac{\mathrm {d}^n}{\mathrm {d}x^n} \left( f(x)- {F}_m[f](x) \right) . \end{aligned}$$

Then the theorem basically says that

$$\begin{aligned} D_{n,m}[f](x) = O\left( h^{r - n + 1/q'} \right) \end{aligned}$$

for functions $f \in L^{r+1}_q([a,b] )$, i.e., the function f is approximated by its $F^m$-transform with an approximation order $O(h^{r+1/q'})$, and its derivatives are also approximated by the derivatives of the $F^m$-transform with a corresponding loss of the approximation order. However, the theorem also provides a tight estimate of the constant hidden in the big-O notation.

Theorem 2

Suppose that nonnegative integers r, m satisfy $r \le \min \left\{ 2m+1,2k-1 \right\}$. Let f be a function from $L^{r+1}_q([a,b] )$ for some $q \ge 1,$ and ${F}_m[f]$ be its integral or discrete $F^m$-transform w.r.t. the (k, N) -FPB of [a, b] (in the discrete case we also require $m \le 2kM-2$ so that $\varDelta$ is sufficiently dense).

Then for all x from the interval $[\hat{a}, \hat{b}]$ (defined by (10)) and $n \in [0 \, ..\, r]$ the estimation

$$\begin{aligned} \left| D_{n,m}[f](x) \right| \le h^{r-n+1/q'} C_{\star } \left\| f^{(r+1) } \right\| _{L^q( \left[ a,b \right] )} , \end{aligned}$$

(19)

holds, where

$$\begin{aligned}&1/q + 1/q' = 1, \\&D_{n,m}[f](x) = \frac{\mathrm {d}^n}{\mathrm {d}x^n} \left( f(x)- {F}_m[f](x) \right) , \\&C_{\star } = \left\| U_{n,m,r,k,u } \right\| _{L^{q'}(\mathcal {I}_k)},\\&j_0 \in [k-1 \, ..\, N-k] \text { is such that }x\in [t_{j_0}, t_{j_0+1}], \\&u = (x-t_{j_0})/h-0.5 , \\&U_{n,m,r,k,u }(\tau )=\frac{\mathrm {d}^n}{\mathrm {d}u^n} K_{r}(u,\tau )-\frac{\mathrm {d}^n}{\mathrm {d}u^n} F_{m,*}[K_{r}(\cdot ,\tau )] (u),\\&K_{r}(s, \tau )=(s-\tau )^r_{+}/r!, \end{aligned}$$

and $F_{m,*}$ is the corresponding (integral or discrete) $F^m$-transform w.r.t. the $(k,2k-1)$-FPB of the fixed interval ${\mathcal {I}_k = [-2k+0.5, 2k-0.5]}$.

Remark 1

The bound (19) is essentially tight: for all $C > 0$ we have

$$\begin{aligned} \sup _{ f \in CW^{r+1}_q([a,b] ) } \left| D_{n,m}[f](x) \right| =C \cdot C_{\star }h^{r-n+1/q'} , \end{aligned}$$

where the set $CW^{r+1}_q([a,b] )$ consists of all functions $f \in L^{r+1}_q([a,b] )$ such that $\left\| f^{(r+1) } \right\| _{L^q([a,b] )} \le C.$

Remark 2

Notice that also in the case of the discrete $F^m$-transform the result (19) is obtained for function f defined on the whole interval [a, b], not only on the discrete grid $\varDelta$.

Proof

We apply for function f Taylor formula with the integral form of the remainder

$$\begin{aligned} f(x) = p_r(x) + \frac{1}{r!} \, \int \limits _{a}^{b} f^{(r+1)} (t) \, (x-t)_+^r \, \,\mathrm {d}t, \end{aligned}$$

where $p_r$ is the Taylor polynomial for f, $\deg p_r \le r$, a is the point of expansion. We rewrite it

$$\begin{aligned} f(x) = p_r(x) + I^r_{f}(x) \end{aligned}$$

using for the remainder $I^r_{f}$ the following form

$$\begin{aligned} I^r_{f}(x) = \int \limits _{a}^{b} f^{(r+1)} (t) \, K_{r}(x,t) \, \,\mathrm {d}t. \end{aligned}$$

(20)

Taking into account that from Theorem 1 it follows that ${F}_m[p_r]\equiv p_r$ on $[\hat{a}, \hat{b}]$, we obtain that for all $x\in [\hat{a}, \hat{b}]$

$$\begin{aligned} D_{0,m}[f](x) = D_{0,m}[I^r_f](x) . \end{aligned}$$

From (20) and Lemma 10 we obtain

$$\begin{aligned} {F}_m[I^r_{f}](x) = \int \limits _{a}^{b} f^{(r+1) }(t) \, {F}_m[K_r( \cdot ,t)](x) \, \,\mathrm {d}t, \end{aligned}$$

therefore for $x \in [\hat{a}, \hat{b}]$ it holds

$$\begin{aligned} D_{0,m}[f](x) =\int \limits _{a}^{b} f^{(r+1) }(t) \, D_{0,m}[K_{r}(\cdot ,t)](x) \mathrm {d}t . \end{aligned}$$

Differentiating this equality n times gives

$$\begin{aligned} D_{n,m}[f](x) =\int \limits _{a}^{b} f^{(r+1) }(t) \, D_{n,m}[K_{r}(\cdot ,t)](x) \mathrm {d}t . \end{aligned}$$

By Hölder’s inequality,

$$\begin{aligned} \left| D_{n,m}[f](x) \right| \le \left\| f^{(r+1) } \right\| _{L^q([a,b])}\, \left\| V_{n,m,r,x} \right\| _{L^{q'}([a,b])}, \end{aligned}$$

where $V_{n,m,r,x}(t) := D_{n,m}[K_{r}(\cdot ,t)](x)$. Moreover, for all $C > 0$ we have

$$\begin{aligned} \sup _{ f \in CW^{r+1}_q([a,b] ) } \left| D_{n,m}[f](x) \right| =C \left\| V_{n,m,r,x} \right\| _{L^{q'}([a,b])}. \end{aligned}$$

Suppose that $x \in \left[ t_{j_0} , t_{j_0+ 1 } \right] \subset [\hat{a}, \hat{b}]$ for some index $j_0 \in [k-1 \, ..\, N-k]$. From Lemma 11 it follows that for all $t \notin \left[ t_{j_0-2k+1} , t_{j_0+ 2k } \right]$ we have

$$\begin{aligned} K_{r}(x ,t) = {F}_m[K_{r}(\cdot ,t)](x) , \end{aligned}$$

i.e., $V_{n,m,r,x}(t)=0$, since

$$\begin{aligned}&K_{r}(x ,t) = {\left\{ \begin{array}{ll} 0,&{} x \le t \\ (x-t)^r/r! , &{} x > t, \end{array}\right. } \\&{F}_m[0] \equiv 0 \text { and } \\&{F}_m[(\cdot -t)^r/r!](x) = (x-t)^r/r! \text {, by Theorem } 1. \end{aligned}$$

Consequently,

$$\begin{aligned} \left\| V_{n,m,r,x} \right\| _{L^{q'}([a,b])}= \left\| V_{n,m,r,x} \right\| _{L^{q'} \left( \left[ t_{j_0-2k+1} , t_{j_0+ 2k } \right] \right) }. \end{aligned}$$

Now let $t \in \left[ t_{j_0-2k+1} , t_{j_0+ 2k } \right]$. Define an affine map $\phi : \left[ t_{j_0-2k+1} , t_{j_0+ 2k } \right] \rightarrow \mathcal {I}_k$ by

$$\begin{aligned} \phi (s) = \frac{s - t_{j_0} }{h}- 0.5 \end{aligned}$$

and denote $u = \phi (x)$, $\tau = \phi (t)$.

Then $K_{r}(x ,t ) = h^{r} K_{r}(u ,\tau )$. From Corollary 3,

$$\begin{aligned} {F}_m[ K_{r}( \cdot ,t ) ](x) = h^r F_{m,*}[K_{r}(\cdot ,\tau )] (u). \end{aligned}$$

Therefore

$$\begin{aligned}&V_{n,m,r,x}(t) =D_{n,m}[K_{r}(\cdot ,t)](x) \\&\quad = \frac{\mathrm {d}^n}{\,\mathrm {d}x^n} \left( K_{r}(x ,t ) - {F}_m[K_r( \cdot ,t)](x) \right) \\&\quad = h^r \frac{\mathrm {d}^n}{\,\mathrm {d}x^n} \left( K_{r}( \phi (x) ,\tau ) - F_{m,*}[K_r( \cdot ,\tau )](\phi (x)) \right) \\&\quad = h^r \frac{\mathrm {d}^n}{\,\mathrm {d}u^n} \left( K_{r}(u ,\tau ) - F_{m,*}[K_r( \cdot ,\tau )](u) \right) \left( \frac{\mathrm {d}\phi (x) }{\,\mathrm {d}x} \right) ^n \\&\quad = h^{r-n} \frac{\mathrm {d}^n}{\,\mathrm {d}u^n} \left( K_{r}(u ,\tau ) - F_{m,*}[K_r( \cdot ,\tau )](u) \right) \\&\quad = h^{r-n} U_{n,m,r,k,u }(\tau ). \end{aligned}$$

Now, if $q'= \infty$, we immediately have

$$\begin{aligned} \left\| V_{n,m,r,x} \right\| _{L^{\infty } ([a,b])}= h^{r-n} \left\| U_{n,m,r,k,u } \right\| _{L^{\infty }( \mathcal {I}_k)}. \end{aligned}$$

Consider the case $q'< \infty$. Then

$$\begin{aligned}&\left\| V_{n,m,r,x} \right\| ^{q'}_{L^{q'}([a,b])}\\&\quad = \int \limits _{t_{j_0-2k+1}}^{t_{j_0+ 2k }} \left| V_{n,m,r,x}(t) \right| ^{q'} \, \,\mathrm {d}t\\&\quad = h^{rq'-nq'+1} \int \limits _{\mathcal {I}_k} \left| U_{n,m,r,k,u }(\tau ) \right| ^{q'} \, \mathrm {d}\tau \\&\quad = h^{(r-n)q'+1} \left\| U_{n,m,r,k,u } \right\| ^{q'}_{L^{q'}(\mathcal {I}_k)}. \end{aligned}$$

We conclude that in both cases

$$\begin{aligned}&\left\| V_{n,m,r,x} \right\| _{L^{q'}([a,b])} \\&\quad = h^{r-n+1/q'} \left\| U_{n,m,r,k,u } \right\| _{L^{q'}( \left[ -2k+0.5,2k -0.5 \right] )}. \end{aligned}$$

The theorem is proven. $\square$

8.2 Numerical examples

For illustrative purposes we consider approximation of functions $f_1(x)= \sin ^2(\pi x)$ and $f_2(x) = \sin (\exp (4x))$ over the interval [0, 1] with the discrete $F^m$-transform, for $m \in \left\{ 0,1 \right\}$ and $N\in \left\{ 12,102 \right\}$ (in the case of $f_2$ we also consider $N\in \left\{ 1002,10002 \right\}$). In these examples the fuzzy partitions will be given by cubic B-splines (i.e., we consider (2, N) -FPB of [a, b] ).

For the sake of comparability, the interval [a, b] is chosen so that [0, 1] is the interval where the Ruspini condition is fulfilled, i.e.,

• when $N=12$, set $a= -0.3$, $b=1.3$, then

  $$\begin{aligned} h=\frac{b-a}{N+2k} =\frac{1.4}{14}=0.1 \end{aligned}$$

  and the Ruspini condition holds on $[\hat{a}, \hat{b}] = [0,1]$;

• when $N=102$, set $a= -0.03$, $b=1.03$, then

  $$\begin{aligned} h=\frac{b-a}{N+2k} =\frac{1.06}{106}=0.01 \end{aligned}$$

  and the Ruspini condition holds on $[\hat{a}, \hat{b}] = [0,1]$.

In either case the parameter M, characterizing the size of the discrete grid (defined in Sect. 3.5), is $M=3$.

For each of the described FPB we consider the error function $\mathcal {E}_{m,N}[f]$, with $m\in \left\{ 0,1 \right\}$, $N\in \left\{ 12,102 \right\}$, defined by

$$\begin{aligned} \mathcal {E}_{m,N}[f](x) = f(x) - {F}_m[f](x), \quad x \in [0,1], \end{aligned}$$

where ${F}_m[f]$ is the $F^m$-transform of f w.r.t. the corresponding FPB and $f \in \left\{ f_1,f_2 \right\}$.

Fig. 4

Approximation error for $f_1$, with parameters $k=2$, $N = 12$, $m=0$ and $M =3$

In Figs. 4 and 5 we plot $\mathcal {E}_{0,12}[f_1]$ and $\mathcal {E}_{0,102}[f_1]$, respectively. In this case Theorem 2 guarantees approximation of order $O(h^{2m+2}) = O(h^2)$, which is consistent with the scaling of the error function when h decreases from 0.1 to 0.01.

In Figs. 6 and 7 we plot $\mathcal {E}_{1,12}[f_1]$ and $\mathcal {E}_{1,102}[f_1]$, respectively, i.e., now we use the first-degree F-transform instead of the ordinary $F^0$-transform. In this case the order of approximation is $O(h^4)$, which is again consistent with the scaling of the error function.

It can be noticed that the error in Figs. 6 and 7 oscillates; despite this phenomenon, it can be verified that the derivative of ${F}_m[f_1]$ is a smooth function approximating the derivative of $f_1$. Likewise, also the second and third derivative of ${F}_m[f_1]$ approximate the respective derivatives of $f_1$, as predicted by Theorem 2.

Fig. 5

Approximation error for $f_1$, with parameters $k=2$, $N = 102$, $m=0$ and $M =3$

Fig. 6

Approximation error for $f_1$, with parameters $k=2$, $N = 12$, $m=1$ and $M =3$

Fig. 7

Approximation error for $f_1$, with parameters $k=2$, $N = 102$, $m=1$ and $M =3$

Now let us consider approximation of the function $f_2(x) = \sin (\exp (4x))$.

In Figs. 8 and 10 we plot $\mathcal {E}_{0,12}[f_2]$ and $\mathcal {E}_{1,12}[f_2]$, respectively. As it can be observed, while the function is well approximated near $x=0$, both $F^0$-transform and $F^1$-transform fail to provide meaningful approximation of $f_2$ near $x=1$ (and, in fact, increasing $m=0$ to $m=1$ yields no advantage when $h=0.1$).

Fig. 8

Approximation error for $f_2$, with parameters $k=2$, $N = 12$, $m=0$ and $M =3$

Consider Figs. 9 and 11, where we plot $\mathcal {E}_{0,102}[f_2]$ and $\mathcal {E}_{1,102}[f_2]$, i.e., now $h=0.01$. It becomes apparent that approximation by the $F^1$-transform is superior to that by the $F^0$-transform. However, the approximation error is still relatively large and the approximation order $O(h^4)$ in the case of $F^1$-transform is not evident. The reason of that is that the constant hidden in the big-O notation for $O(h^4)$ is large in this example and one has to consider larger values of N to observe the asymptotic behavior of the error.

To illustrate the asymptotic behavior of the error $\mathcal {E}_{1,N}[f_2]$, we also plot $\mathcal {E}_{1,1002}[f_2]$ and $\mathcal {E}_{1,10002}[f_2]$ (Figs. 12, 13, respectively). In these cases the step size is $h=10^{-3}$ and $h=10^{-4}$, respectively. As before, the endpoints of the interval [a, b] are chosen so that the Ruspini condition holds on the unit interval [0, 1] .

Now it can be observed that the error rate indeed decreases proportionally to $h^4$, as decreasing h from $10^{-2}$ to $10^{-3}$ decreases the magnitude of the error from ${\approx } 0.5 \times 10^0$ to ${\approx } 10^{-4}$. Likewise, decreasing h from $10^{-3}$ to $10^{-4}$ decreases the magnitude of the error further to ${\approx } 10^{-8}$.

Fig. 9

Approximation error for $f_2$, with parameters $k=2$, $N = 102$, $m=0$ and $M =3$

Fig. 10

Approximation error for $f_2$, with parameters $k=2$, $N = 12$, $m=1$ and $M =3$

Fig. 11

Approximation error for $f_2$, with parameters $k=2$, $N = 102$, $m=1$ and $M =3$

Fig. 12

Approximation error for $f_2$, with parameters $k=2$, $N = 10^3+2$, $m=1$ and $M =3$

Fig. 13

Approximation error for $f_2$, with parameters $k=2$, $N = 10^4+2$, $m=1$ and $M =3$

9 Proofs of the technical claims

9.1 General results

9.1.1 Central difference operator

By $\nabla$ we will denote the central difference operator, satisfying $\nabla f(t) = f(t+0.5) - f(t-0.5)$ for any function f defined on $[t-0.5,t+0.5]\subset \mathbb {R}$; notation $\nabla ^j$ stands for the jth iteration of the operator $\nabla$, i.e., $\nabla ^j f(t) = \nabla (\nabla ( \ldots \nabla f(t)))$.

This operator is linked to the derivatives of the central B-spline and a related construction, polynomials $R_n$, defined in Sect. 9.2 (see Lemmas 15 and 16). This link will be used in the proof of Lemma 18, which is an important auxiliary result for the proof of Lemma 4.

It is easy to see that the central difference operator has the following property:

Lemma 13

Suppose that $f,g : \mathbb {R} \rightarrow \mathbb {R}$ are such functions that for all $t \in \mathbb {R}$ the integral $\int _{\mathbb {R}}f(x)g(x+t)\,\mathrm {d}x$ exists and has a finite value.

Then for all integers $n \ge 0$ we have

$$\begin{aligned} \int \limits _{\mathbb {R}} f(x) \nabla ^n g(x) \,\mathrm {d}x= (-1)^n\int \limits _{\mathbb {R}} \nabla ^n f(x) g(x) \,\mathrm {d}x. \end{aligned}$$

Proof

When $n=0$, the claim trivially holds. Suppose it holds for an integer n, then by the inductive hypothesis we have

$$\begin{aligned} (-1)^n \int \limits _{\mathbb {R}} f(x) \nabla ^{n+1} g(x) \,\mathrm {d}x= \int \limits _{\mathbb {R}} \nabla ^n f(x) \nabla g(x) \,\mathrm {d}x. \end{aligned}$$

Rewriting $\nabla g(x)= g(x+0.5)- g(x-0.5)$ and changing the integration variable yields

$$\begin{aligned}&\int \limits _{\mathbb {R}} \nabla ^n f(x) \nabla g(x) \,\mathrm {d}x\\&\quad =\int \limits _{\mathbb {R}} \nabla ^n f(x) g(x+0.5) \,\mathrm {d}x-\int \limits _{\mathbb {R}} \nabla ^n f(x) g(x-0.5) \,\mathrm {d}x\\&\quad =\int \limits _{\mathbb {R}} \nabla ^n f(x-0.5) g(x) \,\mathrm {d}x-\int \limits _{\mathbb {R}} \nabla ^n f(x+0.5) g(x) \,\mathrm {d}x\\&\quad =- \int \limits _{\mathbb {R}} \nabla ^{n+1}f(x) g(x) \,\mathrm {d}x\end{aligned}$$

and the claim follows. $\square$

9.1.2 A basis of the space $\mathbb {P}_n$

Another general claim shows a way to construct a basis of the space $\mathbb {P}_n$. We will use this fact in the proof of Lemma 19, which is a prerequisite for Lemma 4.

Lemma 14

Suppose that $n \in \mathbb {N}$ and a polynomial p has degree n. Let $x_0$, ..., $x_n \in \mathbb {R}$ be distinct numbers. Then the polynomials $q_j$, defined by $q_j(x) = p(x+x_j)$, $j \in [0 \, ..\, n]$, form a basis of the space $\mathbb {P}_n$.

Proof

First we note that the polynomials $p_j$, defined by $p_j = {p^{(j)}}/{j!}$, $j \in [0 \, ..\, n]$, form a basis of $\mathbb {P}_n$, since $\deg p_j =n-j$. Let us express each polynomial $q_j$ as the unique linear combination in this basis, obtaining the basis transformation matrix. Then it will be necessary and sufficient to show that this matrix is non-singular.

Apply Taylor formula for $q_j(x) = p(x+x_j)$ with x being the point of expansion, then we see that

$$\begin{aligned} p(x+x_j) = \sum _{l=0}^{n}\frac{x_j^l}{l!} p^{(l)}(x) , \end{aligned}$$

i.e.,

$$\begin{aligned} q_j = \sum _{l=0}^{n} x_j^l p_l. \end{aligned}$$

We conclude that the coefficient matrix is

$$\begin{aligned} \begin{pmatrix} 1 &{}\quad x_0 &{}\quad x_0^2 &{}\quad \ldots &{}\quad x_0^n \\ 1 &{}\quad x_1 &{}\quad x_1^2 &{}\quad \ldots &{}\quad x_1^n \\ &{} &{} &{}\quad \ddots \\ 1 &{}\quad x_n &{}\quad x_n^2 &{}\quad \ldots &{}\quad x_n^n \end{pmatrix}, \end{aligned}$$

which is the Vandermonde matrix and is non-singular, since $x_0$, ..., $x_n \in \mathbb {R}$ are distinct. $\square$

9.1.3 Proof of Lemma 3

Let $P^l = p^l_0$, $l \in [0 \, ..\, r]$, then $\deg P^l = l$ and

$$\begin{aligned} p^l_i(x) = P^l(x-i), \quad x \in \mathbb {R} ,\ i \in [0 \, ..\, N]. \end{aligned}$$

Therefore we have

$$\begin{aligned} f(x) = \sum _{l=0}^{r} a_{i,l} P^l (x-i) , \quad i \in [0 \, ..\, N], \end{aligned}$$

or, equivalently,

$$\begin{aligned} \sum _{l=0}^{r} a_{0,l} P^l (x) = \sum _{l=0}^{r} a_{i,l} P^l (x-i) , \quad i \in [0 \, ..\, N]. \end{aligned}$$

Replacing x by $x+i$ in this equality yields

$$\begin{aligned} \sum _{l=0}^{r} a_{0,l} P^l (x+i) = \sum _{l=0}^{r} a_{i,l} P^l (x) , \quad i \in [0 \, ..\, N]. \end{aligned}$$

(21)

Let us show that for each $l \in [0 \, ..\, r]$ and $s \in [0 \, ..\, l]$ there exists a polynomial $\alpha _{l,s} \in \mathbb {P}_{l-s}$ s.t.

$$\begin{aligned} P^l (x+ y) = \sum _{s=0}^{l} \alpha _{l,s}(y) P^s(x) , \quad x,y \in \mathbb {R}. \end{aligned}$$

(22)

Then it will follow that (21) can be rewritten as

$$\begin{aligned} \sum _{l=0}^{r}a_{i,l} P^l (x) = \sum _{l=0}^{r} a_{0,l} \sum _{s=0}^{l} \alpha _{l,s}(i) P^s(x), \end{aligned}$$

or, equivalently,

$$\begin{aligned} \sum _{s=0}^{r} a_{i,s} P^s (x) = \sum _{s=0}^{r} \sum _{l=s}^{r} a_{0,l} \alpha _{l,s}(i) P^s(x), \end{aligned}$$

or

$$\begin{aligned} \sum _{s=0}^{r} \left( a_{i,s} - \sum _{l=s}^{r}a_{0,l} \alpha _{l,s}(i) \right) P^s(x) = 0. \end{aligned}$$

Since this holds for all $x \in \mathbb {R}$ and the polynomials $P^0, \ldots , P^r$ form a basis of $\mathbb {P}_{r}$, it follows that

$$\begin{aligned} a_{i,s} = \sum _{l=s}^{r} a_{0,l} \alpha _{l,s}(i) , \quad s \in [0 \, ..\, r] ,\ i \in [0 \, ..\, N] . \end{aligned}$$

Let $q_{s}$ be defined with

$$\begin{aligned} q_{s}(x) = \sum _{l=s }^r a_{0,l} \alpha _{l,s}(x) , \end{aligned}$$

then $a_{i,s} = q_{s}(i)$. Moreover, since $\deg \alpha _{l,s} \le l-s$, we see that $q_{s} \in \mathbb {P}_{r-s}$.

It remains to prove the existence of $\alpha _{l,s}$ such that (22) holds. Define constants $\gamma _{l,j,s}$ so that the jth derivative of $P^{l}$, $j\in [0 \, ..\, l]$, is expressed as

$$\begin{aligned} \left( P^l \right) ^{(j)} = j! \, \sum _{s=0}^{l-j} \gamma _{l,j,s} P^s \end{aligned}$$

(this is possible since $P^0, \ldots , P^r$ form a basis of $\mathbb {P}_{r}$). For any $x,y \in \mathbb {R}$ apply Taylor formula for $P^l(x+y)$ with x being the point of expansion, obtaining

$$\begin{aligned} P^l(x+y) = \sum _{j=0}^{l} \frac{y^j}{j!}\left( P^l \right) ^{(j)} (x) = \sum _{j=0}^{l} \sum _{s=0}^{l-j}y^j \, \gamma _{l,j,s} P^s(x). \end{aligned}$$

By interchanging the order of summation we obtain

$$\begin{aligned} P^l(x+y) = \sum _{s=0}^l \sum _{j=0}^{l-s} \gamma _{l,j,s} \, y^j \, P^s(x). \end{aligned}$$

Clearly, the polynomial $\alpha _{l,s}(y) = \sum _{j=0}^{l-s}\gamma _{l,j,s} \, y^j$ satisfies the necessary properties. $\square$

9.2 Claims related to the B-splines

Let us note that the derivatives of the central B-splines are closely related to the difference operator:

Lemma 15

$$\begin{aligned} \phi ^{(j)}_n(t) = \nabla ^j\, \phi _{n-j} (t), \quad j \in [0 \, ..\, n-1], \end{aligned}$$

where $\phi _n$ stands for the central B-spline of degree n (defined in Sect. 2.6) and $\nabla$ is the central difference operator (see Sect. 9.1).

Proof

The case $j=0$ is trivial; see Schoenberg (1973, Eq. 1.10) for $j=1$. Suppose that we have

$$\begin{aligned} \phi ^{(j-1)}_n(t) = \nabla ^{j-1}\, \phi _{n-j+1}(t) \end{aligned}$$

for some $j \in [1 \, ..\, n-1]$. Then

$$\begin{aligned}&\phi _n^{(j)}(t) = \left( \phi ^{(j-1)}_n \right) '(t)\\&\quad =\left( \nabla ^{j-1} \phi _{n-j+1} \right) ' (t) =\nabla ^{j-1} \left( \phi _{n-j+1}' (t) \right) \\&\quad =\nabla ^{j-1} \left( \nabla \phi _{n-j} (t) \right) , \end{aligned}$$

and the claim follows. $\square$

Introduce polynomials $R_n$ (similarly defined polynomials are called dual polynomials in Lyche and Mørken (2008)), where $R_0 \equiv 1$ and for $n \ge 1$

$$\begin{aligned} R_n (x) = {\left\{ \begin{array}{ll} \displaystyle x \, \prod \limits _{j = 1}^{ \frac{n-1}{2} } \left( x^2 - j^2 \right) , &{} n \text { is odd;}\\ \displaystyle \prod \limits _{j = 1}^{\frac{n}{2} } \left( x^2 - (j-0.5)^2 \right) , &{} n \text { is even.} \end{array}\right. } \end{aligned}$$

Notice that the polynomials $R_n$ satisfy the following recurrence formulas:

$$\begin{aligned}&R_n(x-0.5) = \left( x- \frac{n}{2} \right) R_{n-1}(x), \end{aligned}$$

(23)

$$\begin{aligned}&R_n(x+0.5) = \left( x+ \frac{n}{2} \right) R_{n-1}(x). \end{aligned}$$

(24)

One can show a property, similar to Lemma 15:

Lemma 16

$$\begin{aligned} \nabla ^j\, R_{n} (x) = \frac{n!}{(n-j)!} \, R_{n-j}(x) , \quad j \in [0 \, ..\, n] . \end{aligned}$$

Proof

We start with

$$\begin{aligned} \nabla R_{n}(x)= & {} R_n(x+0.5) - R_n (x-0.5) \\ \quad&\overset{(23),(24)}{=}&\left( x+ \frac{n}{2} \right) R_{n-1} \left( x \right) - \left( x- \frac{n}{2} \right) R_{n-1}\left( x \right) \\ \quad= & {} n \cdot R_{n-1}(x). \end{aligned}$$

Suppose that we have

$$\begin{aligned} \nabla ^{j-1}\, R_{n}(x) = \frac{n!}{(n+1-j)!} \, R_{n+1-j}(x) \end{aligned}$$

for some $j \in [2 \, ..\, n]$. Then

$$\begin{aligned}&\nabla ^j\, R_{n}(x) =\nabla \left( \nabla ^{j-1}\, R_{n}(x) \right) \\&\quad =\nabla \left( \frac{n!}{(n+1-j)!} \, R_{n+1-j}(x) \right) \\&\quad = \frac{n!}{(n+1-j)!} \, \nabla R_{n+1-j}(x) =\frac{n!}{(n-j)!} \, R_{n-j}(x), \end{aligned}$$

which completes the inductive step. $\square$

Finally, we have that polynomials $R_n$ are orthogonal (w.r.t. the standard $L_2$ inner product) to the central B-spline $\phi _n$, i.e.,

Lemma 17

$$\begin{aligned} \int \limits _{\mathbb {R}} R_{n}(x) \, \phi _{n}(x) \,\mathrm {d}x= 0 \quad \text {for all } n \in \mathbb {N}. \end{aligned}$$

Proof

From (6) we see that

$$\begin{aligned} 4n \int \limits _{\mathbb {R}} R_{n}(x) \, \phi _{n}(x) \,\mathrm {d}x= I_1 + I_2, \end{aligned}$$

where

$$\begin{aligned}&I_1 = 2\int \limits _{\mathbb {R}} (n+1 + 2 x)\phi _{n-1}(x+0.5) R_{n}(x) \,\mathrm {d}x,\\&I_2 =2 \int \limits _{\mathbb {R}} (n+1 - 2 x)\phi _{n-1}(x-0.5) R_{n}(x) \,\mathrm {d}x. \end{aligned}$$

By changing variables, we have

$$\begin{aligned}&I_1 = 2\int \limits _{\mathbb {R}} (n+2 x)\phi _{n-1}(x) R_{n}(x-0.5) \,\mathrm {d}x,\\&I_2 = 2\int \limits _{\mathbb {R}} (n - 2 x)\phi _{n-1}(x) R_{n}(x+0.5) \,\mathrm {d}x. \end{aligned}$$

Applying (23) and (24) yields

$$\begin{aligned}&I_1 = \int \limits _{\mathbb {R}} (n+2 x )(2x-n)\phi _{n-1}(x) R_{n-1}(x) \,\mathrm {d}x,\\&I_2 = \int \limits _{\mathbb {R}} (n-2x)(2x+n)\phi _{n-1}(x) R_{n-1}(x) \,\mathrm {d}x. \end{aligned}$$

Now it is obvious that $I_1 + I_2 = 0$. $\square$

9.3 Claims related to the FPB

9.3.1 Auxiliary results for the proof of Lemma 4

To derive Lemma 4, we need a few auxiliary results; the next lemma shows that the convolution of the central B-spline $\phi _{2k-1}$ and the corresponding “dual” polynomial $R_{2k-1}$ is the polynomial $y^{2k-1}$:

Lemma 18

Denote $A = \phi _{2k-1}$ and $\rho = R_{2k-1}$, i.e.,

$$\begin{aligned} \rho (x) = x\left( x^2 -1^2 \right) \left( x^2 - 2^2 \right) \ldots \left( x^2 - (k-1)^2 \right) . \end{aligned}$$

Then for all $y \in \mathbb {R}$ we have

$$\begin{aligned} \int \limits _{\mathbb {R}} \rho (y-x) A(x) \,\mathrm {d}x= y^{2k-1}. \end{aligned}$$

Proof

It suffices to show that for all integers $j \ge 0$ it holds that

where $\delta _{ji}$ stands for the Kronecker delta, or, equivalently,

$$\begin{aligned} \int \limits _{\mathbb {R}} \rho ^{(j)} (x) A(x) \,\mathrm {d}x= (2k-1)! \, \delta _{j\,(2k-1)}. \end{aligned}$$

The equality is obviously true for $j \ge 2k-1$. Fix any $j \in [0 \, ..\, 2k-2]$, then it remains to show that

$$\begin{aligned} \int \limits _{\mathbb {R}} \rho ^{(j)} (x) A(x) \,\mathrm {d}x= 0. \end{aligned}$$

(25)

It can be shown (by partially integrating and noting that ${{\mathrm{supp}}}A =(-k,k)$) that the left side of (25) equals $(-1)^j \, \int \limits _{\mathbb {R}} \rho (x) A^{(j)} (x) \,\mathrm {d}x$, thus

$$\begin{aligned}&\int \limits _{\mathbb {R}} \rho ^{(j)} (x) A(x) \,\mathrm {d}x= (-1)^j \int \limits _{\mathbb {R}} \rho (x) A^{(j)} (x) \,\mathrm {d}x\\&\quad = (-1)^j \int \limits _{\mathbb {R}} \rho (x) A^{(j)}(x) \,\mathrm {d}x\\&\quad \overset{\text {L.15}}{=} (-1)^j \int \limits _{\mathbb {R}} \rho (x) \nabla ^j\, \phi _{2k-1-j} (x) \,\mathrm {d}x\\&\quad \overset{}{=} \int \limits _{\mathbb {R}} \nabla ^j\, \rho (x) \, \phi _{2k-1-j}(x) \,\mathrm {d}x\\&\quad \overset{\text {L.16}}{=} \frac{(2k-1)!}{(2k-1-j)!} \int \limits _{\mathbb {R}} R_{2k-1-j} (x) \, \phi _{2k-1-j}(x) \,\mathrm {d}x=0, \end{aligned}$$

where the last equality follows from Lemma 17. Therefore (25) holds, which completes the proof. $\square$

Next we show a result which relates the sum in Lemma 4 with an integral of $f \cdot A$; as before, A stands for the central B-spline of degree $2k-1$ and $\rho$ is the corresponding ’dual’ polynomial $R_{2k-1}$.

Lemma 19

For all $\tau \in [0,1]$ and $f \in \mathbb {P}_{2k-1}$ it holds that

$$\begin{aligned} \int \limits _{-k}^k f(x)A(x)\,\mathrm {d}x= \sum _{j=1-k}^{k} f(j -\tau ) A(j-\tau ). \end{aligned}$$

Proof

Fix any $\tau \in [0,1]$. Setting $n=2k-1$, $x=\tau$ and

$$\begin{aligned} t_i = -n, \ j=i+n , \ t_{l+1} =t_l +1 \text { for all }l \in [i \, ..\, j], \end{aligned}$$

in the Marsden’s identity (4) yields

$$\begin{aligned} (z-\tau )^{2k-1}=\sum _{j=1-k}^{k} \rho (z -j) A(\tau -j ) , \quad z \in \mathbb {R} . \end{aligned}$$

Now let $z=y+ \tau$ for any $y \in \mathbb {R}$, obtaining

$$\begin{aligned} y^{2k-1} = \sum _{j=1-k}^{k} \rho (y-j+\tau ) A(\tau -j). \end{aligned}$$

On the other hand, it follows from Lemma 18 that for all $y \in \mathbb {R}$ we have

$$\begin{aligned} \int \limits _{-k}^k \rho (y-x)A(x)\,\mathrm {d}x= y^{2k-1} . \end{aligned}$$

The two equalities (and the fact that A is an even and $\rho$ is an odd function) yield

$$\begin{aligned} \int \limits _{-k}^k \rho (x-y)A(x)\,\mathrm {d}x= \sum _{j=1-k}^{k} \rho (j-\tau -y) A(j-\tau ) \end{aligned}$$

for all $y \in \mathbb {R}$. In particular, setting $y=-i$, $i \in [-k \, ..\, k]$, gives us

$$\begin{aligned} \int \limits _{-k}^k \rho (x+i)A(x)\,\mathrm {d}x= \sum _{j=1-k}^{k} \rho (j-\tau + i) A(j-\tau ). \end{aligned}$$

(26)

Suppose that $\displaystyle f \in \mathbb {P}_{2k-1}$. Since $\deg \rho = 2k-1$, by Lemma 14 there exist real numbers $f_i$, $i \in [-k \, ..\, k]$, s.t.

$$\begin{aligned} f(x) = \sum _{i=-k}^{k} f_i \rho (x+i) \quad \text {for all }x \in \mathbb {R}. \end{aligned}$$

(27)

Now we obtain

$$\begin{aligned}&\int \limits _{-k}^k f(x)A(x)\,\mathrm {d}x\\&\quad \overset{(27)}{=} \sum _{i=-k}^{k} f_i \int \limits _{-k}^k \rho (x+i)A(x)\,\mathrm {d}x\\&\quad \overset{(26)}{=}\sum _{i=-k}^{k} f_i \sum _{j=1-k}^{k} \rho (j-\tau + i) A(j-\tau ) \\&\quad \overset{(27)}{=} \sum _{j=1-k}^{k} f(j-\tau ) A(j-\tau ). \end{aligned}$$

Thus

$$\begin{aligned} \int \limits _{-k}^k f(x)A(x)\,\mathrm {d}x= \sum _{j=1-k}^{k} f(j-\tau ) A(j-\tau ) \end{aligned}$$

(28)

for all $f \in \mathbb {P}_{2k-1}$ and $\tau \in [0,1]$.$\square$

Now Lemma 4 easily follows from Lemma 19.

9.3.2 Proof of Lemma 4

Fix a number $\tau \in [0,1]$ and a polynomial p such that its degree satisfies $\deg p \le \min \left\{ l-1, 2k-1-l \right\}$. Set $f(x) = p(x + \tau ) P(x)$, then $f \in \mathbb {P}_{2k-1}$ and from Lemma 19 we obtain

$$\begin{aligned} \int \limits _{-k}^k p(x + \tau ) P(x) A(x)\,\mathrm {d}x= \sum _{j=1-k}^{k} p(j) P(j-\tau ) A(j-\tau ). \end{aligned}$$

Now the left side equals 0 (due to the first requirement on P in the statement of Lemma) and we conclude

$$\begin{aligned} \sum _{j=1-k}^{k} p(j) P(j-\tau ) A(j-\tau ) = 0. \end{aligned}$$

Taking into account that P is either an odd or an even function and A is even completes the proof. $\square$

9.4 Claims related to the template spaces

Lemma 6 simply states that the integral $\int _{-k} ^k p(t) A(t) \,\mathrm {d}t$ can be well approximated with the corresponding sum over the points in the template $Z$ (defined in Sect. 4.3.1), in the sense that this approximation is precise for all polynomials $p \in \mathbb {P}_{2k-1}$. To show that, we invoke once more Lemma 19.

9.4.1 Proof of Lemma 6

Notice that $A(k) = 0$, hence we can rewrite the desired equality as

$$\begin{aligned} \int \limits _{-k} ^k p(t) A(t) \,\mathrm {d}t= \frac{1}{M} \sum _{j=-Mk+1}^{Mk} p \left( j/M \right) A\left( j/M \right) , \end{aligned}$$

or, equivalently,

$$\begin{aligned} \int \limits _{-k} ^k p(t) A(t) \,\mathrm {d}t= \frac{1}{M} \sum _{s=0}^{M-1} \sigma _s, \end{aligned}$$

(29)

where

$$\begin{aligned} \sigma _s := \sum _{j=-k+1}^{k} p \left( j - \frac{s}{M} \right) A \left( j - \frac{s}{M} \right) , \quad s\in [0 \, ..\, M-1] . \end{aligned}$$

Since $p \in \mathbb {P}_{2k-1}$ and for all $s\in [0 \, ..\, M-1]$ we have $s/M \in [0,1]$, from Lemma 19 it follows that

$$\begin{aligned} \sigma _s = \int \limits _{-k}^k p(x)A(x)\,\mathrm {d}x, \text { for all } s\in [0 \, ..\, M-1] , \end{aligned}$$

thus also

$$\begin{aligned} \frac{1}{M} \sum _{s=0}^{M-1} \sigma _s =\int \limits _{-k}^k p(x)A(x)\,\mathrm {d}x. \end{aligned}$$

Hence (29) holds, concluding the proof. $\square$

9.4.2 Proof of Lemma 5

We have

$$\begin{aligned} \left\langle f_i,g_i \right\rangle _{i}= \sum _{z \in \varDelta _i } f_i(z)g_i(z) A_i(z). \end{aligned}$$

Since $z \in \varDelta _i$ if and only if $\tilde{z} = \frac{z-t_i}{h}$ is in the set $Z$, rewrite

$$\begin{aligned}&f_i(z) = f(\tilde{z}), \\&g_i(z) = g(\tilde{z}), \\&A_i(z) = A(\tilde{z}), \end{aligned}$$

for all $z \in \varDelta _i$, $\tilde{z} = \frac{z-t_i}{h} \in Z$. Then

$$\begin{aligned}&\left\langle f_i,g_i \right\rangle _{i}= \sum _{z \in \varDelta _i } f_i(z)g_i(z) A_i(z) \\&\quad = \sum _{\tilde{z} \in Z} f(\tilde{z})g (\tilde{z}) A (\tilde{z}) = \left\langle f, g \right\rangle , \end{aligned}$$

and the claim follows. $\square$

9.5 Claims related to the FPB-based discrete transform

9.5.1 Auxiliary results for the proof of Lemmas 8 and 9

We start with an auxiliary lemma which allows to describe matrices $\mathbf {X}_i^T \mathbf {A}_i \mathbf {X}_i$ and $\mathbf {X}_i^T \mathbf {A}_i \mathbf {Y}$ in terms of arbitrary polynomials, spanning $\mathbb {P}_m$.

Lemma 20

Let $m \ge 0$ and matrices $\mathbf {X}_i$, $\mathbf {A}_i$, $\mathbf {Y}$ be defined as in Definition 5. Suppose that polynomials $p^0$, $p^1$, ..., $p^m$ span the space $\mathbb {P}_m$.

Let $\mathbf {P}$ be the invertible basis transformation matrix satisfying

$$\begin{aligned} \mathbf {P} \begin{pmatrix} 1 \\ t \\ \vdots \\ t^m \end{pmatrix} = \begin{pmatrix} p^0(t) \\ p^1(t) \\ \vdots \\ p^m(t) \end{pmatrix}, \quad \text { for all }t \in \mathbb {R}. \end{aligned}$$

Define also matrices

$$\begin{aligned}&\mathbf {X} = \begin{pmatrix} p^l(\tilde{z}_{r}) \end{pmatrix}_{r,l}, \ r \in [-Mk+1,Mk-1], \ l \in [0 \, ..\, m]; \\&\mathbf {A} = {{\mathrm{diag}}}\left( A(\tilde{z}_{-Mk+1}), \ldots , A(\tilde{z}_{Mk-1}) \right) ; \\&\mathbf {D}_h = {{\mathrm{diag}}}(1,h,h^2,\ldots ,h^m). \end{aligned}$$

Then

$$\begin{aligned} \mathbf {X}_i^T \mathbf {A}_i \mathbf {X}_i = \mathbf {D}_h \mathbf {P}^{-1} \mathbf {X}^T \mathbf {A} \mathbf {X} \left( \mathbf {P}^{-1} \right) ^T \mathbf {D}_h \end{aligned}$$

and

$$\begin{aligned} \mathbf {X}_i^T \mathbf {A}_i \mathbf {Y} = \mathbf {D}_h \mathbf {P}^{-1} \mathbf {X}^T \mathbf {A} \mathbf {Y}_i . \end{aligned}$$

Proof

Let $q^l(t) = t^l$, $l \in [0 \, ..\, m]$, and $\hat{\mathbf {X}}$ be the matrix with entries $\hat{X}_{r,l} = (\tilde{z}_{r})^l$ (i.e., $\hat{\mathbf {X}}$ coincides with the matrix $\mathbf {X}$ when $p^l \equiv q^l$ for all l).

Notice that $\mathbf {A}_i$ has zero rows/columns when the corresponding entry $A_i(z_{j})$ is zero, which happens whenever $j \notin \varDelta _i$. Remove all rows and columns of $\mathbf {A}_i$ which correspond to $A_i(z_{j})$ with $j \notin \varDelta _i$ and all corresponding rows in $\mathbf {X}_i$ and $\mathbf {Y}$. Denote the obtained matrices by $\tilde{{ \mathbf {A}}}_i$, $\tilde{{\mathbf {X}}}_i$ and $\mathbf {Y}_i$ (notice that we obtain the same $\mathbf {Y}_i$ which was defined in the statement of the Lemma). Then

$$\begin{aligned} \mathbf {X}_i^T \mathbf {A}_i \mathbf {X}_i = \tilde{{\mathbf {X}}}_i^T \tilde{{\mathbf {A}}}_i \tilde{\mathbf {X}}_i \end{aligned}$$

and

$$\begin{aligned} \mathbf {X}_i^T \mathbf {A}_i \mathbf {Y} = \tilde{{\mathbf {X}}}_i^T \tilde{{\mathbf {A}}}_i \mathbf {Y}_i. \end{aligned}$$

The matrix $\tilde{{\mathbf {A}}}_i$ is the diagonal matrix with entries $A_i(z)$, $z \in \varDelta _i$ on its diagonal. However, these values coincide with $A(\tilde{z})$, where $\tilde{z} = \frac{z-t_i}{h} \in Z$. Thus $\tilde{{\mathbf {A}}}_i = \mathbf {A}$. Moreover, for all $z \in \varDelta _i$ we have

$$\begin{aligned} \left( z - t_i \right) ^l = \left( \tilde{z} h \right) ^l = q^l(\tilde{z} ) \cdot h^l. \end{aligned}$$

Hence lth column of $\tilde{{\mathbf {X}}}_i$ is obtained as the corresponding column of $\hat{\mathbf {X}}$, multiplied with $h^l$, i.e.,

$$\begin{aligned} \tilde{{\mathbf {X}}}_i = \hat{\mathbf {X}} \cdot \mathbf {D}_h . \end{aligned}$$

Therefore

$$\begin{aligned} \mathbf {X}_i^T \mathbf {A}_i \mathbf {X}_i = \mathbf {D}_h \hat{\mathbf {X}} ^T \mathbf {A} \hat{\mathbf {X}} \mathbf {D}_h \end{aligned}$$

(30)

and

$$\begin{aligned} \mathbf {X}_i^T \mathbf {A}_i \mathbf {Y}= \mathbf {D}_h \hat{\mathbf {X}} ^T \mathbf {A} \mathbf {Y}_i. \end{aligned}$$

(31)

Finally, the basis transformation matrix $\mathbf {P}$ satisfies

$$\begin{aligned} \hat{\mathbf {X}} = \mathbf {X} \left( \mathbf {P}^{-1} \right) ^T. \end{aligned}$$

Combining this with (30) and (31) yields the desired equalities. $\square$

9.5.2 Proof of Lemma 8

The discrete grid $\varDelta$ is sufficiently dense iff $\mathbf {X}_i^T \mathbf {A}_i \mathbf {X}_i$ is non-singular for each i. We apply Lemma 20 with $p^l = \mathtt {P}_{l}$, then $\mathbf {X}_i^T \mathbf {A}_i \mathbf {X}_i$ is non-singular iff $\mathbf {X}^T \mathbf {A} \mathbf {X}$ is non-singular.

Notice that the (i, j) th entry of $\mathbf {X}^T \mathbf {A} \mathbf {X}$ equals

$$\begin{aligned} \sum _{r=-Mk+1}^{Mk-1} \mathtt {P}_{i} (\tilde{z}_{r}) \mathtt {P}_{j}(\tilde{z}_{r}) A(\tilde{z}_{r}) = \left\langle \mathtt {P}_{i}, \mathtt {P}_{j} \right\rangle . \end{aligned}$$

The RHS is nonzero iff $i=j$, i.e., $\mathbf {X}^T \mathbf {A} \mathbf {X}$ is the diagonal matrix with entries $\left\| \mathtt {P}_{l} \right\| ^2$, $l \in [0 \, ..\, m]$, on its diagonal. Since $\left\| \mathtt {P}_{l} \right\| =0$ for all $l \in [0 \, ..\, 2kM-2]$ and $\left\| \mathtt {P}_{l} \right\| =0$ for $l \ge 2kM-1$, the matrix $\mathbf {X}^T \mathbf {A} \mathbf {X}$ is non-singular iff $m \le 2kM-1$.

$\square$

9.5.3 Proof of Lemma 9

Recall that

$$\begin{aligned} F_{m,i}^\rightarrow {}[f](t) = \left( \mathbf {c}^{(i)} \right) ^T \begin{pmatrix} 1 \\ t - t_i \\ \vdots \\ (t-t_i)^m \end{pmatrix}, \end{aligned}$$

where

$$\begin{aligned} \mathbf {c}^{(i)}= \left( \mathbf {X}_i^T \mathbf {A}_i \mathbf {X}_i \right) ^{-1} \mathbf {X}_i^T \mathbf {A}_i \mathbf {Y}. \end{aligned}$$

Apply Lemma 20 to get

$$\begin{aligned}&\mathbf {c}^{(i)}= \\&\quad = \left( \mathbf {D}_h \mathbf {P}^{-1} \mathbf {X}^T \mathbf {A} \mathbf {X} \left( \mathbf {P}^{-1} \right) ^T \mathbf {D}_h \right) ^{-1} \mathbf {D}_h \mathbf {P}^{-1} \mathbf {X}^T \mathbf {A} \mathbf {Y}_i =\\&\quad = \mathbf {D}_h^{-1} \mathbf {P}^ T \left( \mathbf {X}^T \mathbf {A} \mathbf {X} \right) ^{-1} \mathbf {X}^T \mathbf {A} \mathbf {Y}_i = \mathbf {D}_h^{-1}\mathbf {P}^ T \beta ^{(i)} \end{aligned}$$

and

$$\begin{aligned} \left( \mathbf {c}^{(i)} \right) ^T = \left( \beta ^{(i)} \right) ^T \mathbf {P} \mathbf {D}_h^{-1}. \end{aligned}$$

Notice that

$$\begin{aligned} \mathbf {D}_h^{-1} \begin{pmatrix} 1 \\ t - t_i \\ \vdots \\ (t-t_i)^m \end{pmatrix} = \begin{pmatrix} 1 \\ \tilde{t} \\ \vdots \\ \tilde{t}^m \end{pmatrix}, \quad \tilde{t} := \frac{t - t_i}{h}. \end{aligned}$$

On the other hand, since $p^l(\tilde{t}) = p_i^l(t)$, we have

$$\begin{aligned} \mathbf {P} \begin{pmatrix} 1 \\ \tilde{t} \\ \vdots \\ \tilde{t}^m \end{pmatrix} = \begin{pmatrix} p^0(\tilde{t}) \\ p^1(\tilde{t}) \\ \vdots \\ p^m(\tilde{t}) \end{pmatrix} = \begin{pmatrix} p^0_i(t) \\ p^1_i( t) \\ \vdots \\ p^m_i(t) \end{pmatrix}. \end{aligned}$$

It follows that

$$\begin{aligned} F_{m,i}^\rightarrow {}[f](t) = \left( \beta ^{(i)} \right) ^T \begin{pmatrix} p^0_i(t) \\ p^1_i( t) \\ \vdots \\ p^m_i(t) \end{pmatrix} = \sum _{l=0}^{m} \beta ^{(i)}_l p_i^l(\tilde{t}), \end{aligned}$$

which concludes the proof. $\square$

9.6 Additional properties of $F^m$-transforms

9.6.1 Proof of Lemma 10

We prove that for all $l \in [0 \, ..\, m]$ and $i \in [0 \, ..\, N]$ it holds that

$$\begin{aligned} c_{i,l}[f] = \int \limits _{a}^{b} g(t) \, c_{i,l}[K( \cdot ,t)] \, \,\mathrm {d}t, \end{aligned}$$

where $c_{i,l}[f]$ stands for the coefficient at $\mathtt {P}_{i,l}$ of the ith component of either the direct integral ${F}_m^{\mathrm {int}}[f]$ (in this case $c_{i,l}[f]= c_{i,l}^{\mathrm {int}}[f]$) or the direct discrete ${F}_m^{\mathrm {dsc}}[f]$ (in this case $c_{i,l}[f]= c_{i,l}^{\mathrm {dsc}}[f]$) $F^m$-transform of f.

The integral case

$$\begin{aligned} c_{i,l} [f]= & {} \frac{1}{\left\| \mathtt {P}_{i,l} \right\| ^2_i}\, \int \limits _{\mathbb {R}} f(x) \mathtt {P}_{i,l}(x) A_i(x) \,\mathrm {d}x\\= & {} \frac{1}{\left\| \mathtt {P}_{i,l} \right\| ^2_i}\, \int \limits _{a}^{b} g(t) \,\int \limits _{\mathbb {R}} K(x,t) \mathtt {P}_{i,l}(x) A_i(x) \mathrm {d}x \, \,\mathrm {d}t\\= & {} \int \limits _{a}^{b} g(t) \, c_{i,l} [K(\cdot ,t)] \, \,\mathrm {d}t. \end{aligned}$$

The discrete case

$$\begin{aligned} c_{i,l} [f]= & {} \frac{1}{\left\| \mathtt {P}_{i,l} \right\| ^2_i}\, \sum _{z \in \varDelta _i} f(z) \mathtt {P}_{i,l}(z) A_i(z) \\= & {} \frac{1}{\left\| \mathtt {P}_{i,l} \right\| ^2_i}\, \int \limits _{a}^{b} g(t) \, \sum _{z \in \varDelta _i} K(z,t) \mathtt {P}_{i,l}(z) A_i(z) \, \,\mathrm {d}t\\= & {} \int \limits _{a}^{b} g(t) \, c_{i,l} [K(\cdot ,t)] \, \,\mathrm {d}t. \end{aligned}$$

Then for all $x \in [\hat{a}, \hat{b}]$

$$\begin{aligned} {F}_m[f](x)= & {} \sum _{i=0}^{N}\, \sum _{l=0}^{m} c_{i,l}[f] \, \mathtt {P}_{i,l}(x) \, A_i(x) \\= & {} \int \limits _{a}^{b} g(t) \,\sum _{i=0}^{N}\, \sum _{l=0}^{m} c_{i,l}[K( \cdot ,t)] \, P_l^j(x) \, A_i(x) \, \,\mathrm {d}t\\= & {} \int \limits _{a}^{b} g(t) \, {F}_m[K( \cdot ,t)] (x) \, \,\mathrm {d}t, \end{aligned}$$

and the claim follows. $\square$

9.6.2 Proof of Lemma 11

We start with an obvious observation: if $f \equiv g$ on ${{\mathrm{supp}}}A_i$ for some basic function $A_i$, then for all l it holds that $c_{i,l}[f] = c_{i,l}[g]$.

Since ${{\mathrm{supp}}}A_i = \left( t_{i-k}, t_{i+k} \right)$ for all $i\in [0 \, ..\, N]$, it means that in our case

• if $i \le i_0 - k$, then $c_{i,l}[f] =c_{i,l}[g_1]$;

• if $i \ge i_0 + k +1$, then $c_{i,l}[f] = c_{i,l}[g_2]$.

On the other hand, we have

$$\begin{aligned} {F}_m[f](x) = \sum _{i=j_0-k+1}^{j_0+k} \sum _{l=0}^{m} c_{i,l}[f] \mathtt {P}_{i,l}(x) A_i(x), \end{aligned}$$

(32)

since $A_i(x) = 0$ when $i \le j_0-k$ or $i \ge j_0 + k+1$.

It remains to consider the following cases:

• 1. When $j_0 \le i_0 - 2k$, we have $i \le i_0-k$ for all indices $i \in [ j_0-k+1 \, ..\, j_0 + k ]$, thus in (32) all coefficients $c_{i,l}[f]$ satisfy $c_{i,l}[f]= c_{i,l}[g_1]$. Hence it holds that ${F}_m[f](x) ={F}_m[g_1](x)$.

• 2. When $j_0 \ge i_0+2k$, we have $i \ge i_0+k+1$ for all indices $i \in [ j_0-k+1 \, ..\, j_0 + k ]$, thus in (32) all coefficients $c_{i,l}[f]$ satisfy $c_{i,l}[f]= c_{i,l}[g_2]$. Hence ${F}_m[f](x) ={F}_m[g_2](x)$. $\square$

9.6.3 Proof of Lemma 12

First let us show $\tilde{c}_{i,l}[g] = c_{i,l}[f]$, for all $i \in [0 \, ..\, N]$ and $l \in [0 \, ..\, m]$.

The integral case

since $\mathrm {d}(\phi (x)) = \alpha \,\mathrm {d}x$.

The discrete case

$$\begin{aligned} \tilde{c}_{i,l}[g]= & {} \frac{\sum _{u \in \tilde{\varDelta }_i} g(u) \tilde{\mathtt {P}}_{i,l}(u) \tilde{A}_i(u)}{\sum _{u \in \tilde{\varDelta }_i} { \tilde{\mathtt {P}}_{i,l}(u)} ^2 \tilde{A}_i(u) } \\= & {} \frac{\sum _{z \in \varDelta _i} f(z) \mathtt {P}_{i,l}(z) A_i(z)}{\sum _{z \in \varDelta _i} { \mathtt {P}_{i,l}(z)} ^2 A_i(z) } =c_{i,l}[f]. \end{aligned}$$

Now we conclude that for all $u \in [\phi (\hat{a}), \phi (\hat{b})]$

$$\begin{aligned} F_{m,\phi } [g](u)= & {} \sum _{i=0}^{N}\, \sum _{l=0}^{m} \tilde{c}_{i,l}[g] \, \tilde{\mathtt {P}}_{i,l}(u) \tilde{A}_i(u) \\= & {} \sum _{i=0}^{N}\, \sum _{l=0}^{m} c_{i,l}[f] \, \mathtt {P}_{i,l}(x) A_i(x) ={F}_m[f] (x), \end{aligned}$$

where $x:= \phi ^{-1}(u) \in [\hat{a}, \hat{b}]$, which completes the proof.

$\square$

10 Conclusion

The error estimation proved in this paper shows that using B-splines as basic functions of the uniform fuzzy partition for the $F^{m}$-transform allows us to improve results on approximation by the higher-degree continuous and discrete F-transforms. According to our estimation the best result will be achieved by taking B-splines of degree $2m+1$ (or greater) and considering the approximation error for functions from the space $L^{2m+2}_q$. In this case we proved the error bound is characterized with $O(h^{2m+1+ 1/q'})$, where h is the parameter of the uniform partition and $1/q+1/q'=1$. For functions from the space $L^{2m+2}_{\infty }$ (the case $q=\infty$) we obtain $O(h^{2m+2})$. For a comparison let us note that the best result for the approximation error bound, which is known for the $F^{m}$-transform w.r.t. an arbitrary uniform fuzzy partition with parameter h is $O(h^{m+1})$.

Due to a large part of applications of F-transforms in the discrete two-dimensional case, we need to extend the obtained results by considering approximation properties of the composite $F^m$-transform with respect to fuzzy partition given by bivariate central B-splines of odd degree for each variable. In our next paper we will describe and investigate such construction.

Another interesting direction for future research includes applications of B-spline based fuzzy partitions for solving differential equations. Taking into account that in the last years the technique of F-transform has been recognized as a tool to approximate solutions of ordinary and fuzzy differential equations (e.g., Chen and Shen 2014; Khastan et al. 2016; Radi and Stefanini 2015), we hope that the obtained benefits of B-spline based F-transforms in approximation of smooth functions and their derivatives allows us to improve the effectiveness of numerical procedures for differential equations.