On Symmetric Compactly Supported Wavelets with Vanishing Moments Associated to $E_d^{(2)}(\mathbb {Z})$ Dilations

Abstract

Let A be an expansive linear map on ${{\mathbb {R}}}^d$ preserving the integer lattice and with $| \det A|=2$. We prove that if there exists a self-affine tile set associated to A, there exists a compactly supported wavelet with any desired number of vanishing moments and some symmetry. We put emphasis on construction of wavelets associated to a linear map A on ${{\mathbb {R}}}^2$ and to the Quincunx dilation on ${{\mathbb {R}}}^3$ because we can remove the hypothesis of the existence of the self-affine tile set. Our construction is based on low pass filters by Han in dimension one with the dyadic dilation and multiresolution theory. Finally, for some particular dilation matrices, we realize that unidimensional Daubechies low pass filers can be adapted to obtain compactly supported wavelets with any desired degree of regularity and any fix number of vanishing moments.

1 Introduction and Basic Notation

The study of wavelets is attractive from both the theoretical and the applied point of view. That is mainly because wavelets have a simple structure of translations and dilations and they let us give characterizations of function spaces in terms of its coefficients, see e.g. the texts by Daubechies [18], Hernández and Weiss [34] and Wojtaszczyk [56]. In addition, wavelets play an important role in signal analysis, image processing, compression of data and even, in approximation to solutions of differential equations, see e.g. the books by Walnut [55] and Han [33].

In using wavelets, several important points must be taken into considerations. Among them are smoothness, symmetry and number of vanishing moments of those wavelets. Vanishing moments are related to properties of approximation of wavelets because they limit the ability of wavelets to represent polynomials or information in a signal. Many applications of wavelets, for instance, compression of large amount of data, are based on a high number of vanishing moments of a wavelet, see e.g. [9]. Wavelets with smooth properties are important for compression of information, as one can see in Antonini, Barlaud, Mathieu and Dabuechies [1]. Often for its better visual quality and for reducing boundary effects of bounded data, one more desirable property of wavelets is symmetry, see e.g. Han [33].

In dimension one and with dyadic dilation, compactly supported orthonormal wavelets with any number of vanishing moments and any degree of regularity were first constructed by Daubechies [17]. In the same context, the only (up to an integer shift) real valued compactly supported orthogonal wavelet with symmetry is the Haar wavelet, which is also shown in [17]. Antisymmetric compactly supported orthonormal dyadic complex wavelets with high vanishing moments have been obtained by Han [30]. For dilation factors 3 and 4, symmetric and antisymmetric orthogonal wavelets have been constructed by Chui and Lian [16] and by Han [24] respectively. When the dilation factor is larger than 2, a method for constructing symmetric orthogonal scaling functions is suggested by Belogay and Wang [8]. For a low pass filter associated to an orthogonal scaling function with a factor dilation, an algorithm to obtain unitary matrices with symmetry is given by Petukhov [52]. Two families of compactly supported symmetric orthonormal complex M-wavelets with arbitrarily high vanishing moments were presented by Han [29].

Since many problems in applications are multidimensional, many authors are interested in multivariate wavelets. Compactly supported wavelets on $\mathbb {R}^d$ are often constructed by tensor product of univariate wavelets. These types of wavelets are usually called separable wavelets and they have been widely used due to their easy implementation. However, they have some drawbacks, for instance, as Cohen and Daubechies [15] mentions, separable wavelets in $\mathbb {R}^2$ give particular importance to the horizontal and vertical directions. With respect to image processing in $\mathbb {R}^2$, Belogay and Wang [7] pointed out that separable wavelets have very little design freedom, and the product structure can create unpleasant effects for natural images. With the hope to avoid these disadvantages, there is an increasing interest for constructing wavelets by non-tensor product methods. All these considerations show why we are particularly interested in the construction of multivariate compactly supported orthonormal wavelets associated to a general dilation matrix with high vanishing moments, any desired degree of regularity and some symmetry.

In the multivariate context and with the dyadic dilation, several interesting construction of nonseparable compactly supported wavelets appeared in Ayache [2, 3], where the wavelets can be constructed with a high degree of regularity. More constructions of non separable compactly supported wavelets of different nature are given by Peng [51], by Lai [43] and by Karoui [36,37,38]. In these last papers, regularity and accuracy properties of the constructed wavelets are studied.

In the light of the actual state of the art, the problem of constructing compactly supported wavelets associated to a general dilation turns out to be of a different nature. The paper by Gröchenig and Madych [22] is devoted to construct multidimensional Haar type of wavelets with a general dilation. In Lagarias and Wang [42], the existence of Haar wavelets associated to expanding matrices and lattices in $\mathbb {R}^2$ is proved in constructive way using number theory. Bownik [11] proved that this result by Lagarias and Wang can be extended to ${{\mathbb {R}}}^3$.

When the considered dilations have determinant $\pm \,2$, Cohen and Daubechies [15] built two dimensional nonseparable compactly supported wavelets. They focus on the quincunx case and study regularity properties of such wavelets. From a different point of view, another construction of nonseparable wavelets is presented by Kovacevic and Vetterli [39]. More families of compactly supported wavelets with a quincunx matrix are presented in Maass [47]. Belogay and Wang [7] dealt with the dilation given by $A= \left( \begin{array}{cccc} 0 &{} 2 \\ 1 &{} 0 \end{array} \right)$ and emphasis on compactly supported scaling functions and wavelets with high degree of regularity and accuracy. A generalization of these results appeared in Leng, Huang and Cattani [45].

An approach to construct compactly supported wavelets associated to dilations in ${{\mathbb {R}}}^2$ with determinant 4 and 8 is given by Banas [5, 6]. With a general matrix dilation, symmetric refinable scaling functions were studied in Han [25, 28], as well as examples of quincunx wavelets.

As far as the authors know, it is still an open problem the existence of multivariate compactly supported wavelets associated to some expansive linear map, for instance, with the linear map associated to the matrix

$$\begin{aligned} A= \left( \begin{array}{cccccccccc} 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} -1 &{} 2 \\ -1 &{} 0 &{} -1 &{} 1 \end{array} \right) . \end{aligned}$$

Sometimes, the known compactly supported wavelets associated to a general dilation may not have high regularity. This is the case of wavelets associated to the quincunx matrix in ${{\mathbb {R}}}^2$. Let us mention that arbitrarily smooth orthonormal wavelets having exponential decay are constructed in a paper by Han [26].

In this paper, we consider $A: {{\mathbb {R}}}^{d} \rightarrow {{\mathbb {R}}}^d$, a fixed linear map such that $A({{\mathbb {Z}}}^d) \subset {{\mathbb {Z}}}^d$ with $|\det A|=2$, and assume that there is a characteristic function such that it is a scaling function associated to a multiresolution analysis with A. Then, in a constructive way, we show the existence of antisymmetric compactly supported wavelets with any desired number of vanishing moments. We focus on compactly supported wavelets with a linear map A on ${{\mathbb {R}}}^2$ and with the Quincunx matrix on ${{\mathbb {R}}}^3$, because the assumption of the existence of a tile set can be removed. Our construction is based on the structure of the quotient group ${{\mathbb {Z}}}^d/A{{\mathbb {Z}}}^d$ and the general techniques of constructing wavelets from a low pass filter associated to an orthonormal compactly supported scaling function in a multiresolution analysis. We will adapt to the multivariate case, those complex valued low pass filters by Han [30] in dimension one with the dyadic dilation. Finally, we realize that unidimensional Daubechies low pass filers [17] can be adapted to obtain compactly supported wavelets with any desired degree of regularity and any fix number of vanishing moments associated to some special dilation matrices. Furthermore, we give an estimation for both the number of vanishing moments and the degree of regularity of our wavelets. We work in the Fourier transform side.

We begin with notation and definitions. The sets of strictly positive integers, integers, rational numbers, real numbers and complex numbers will be denoted by ${{\mathbb {N}}}$, ${{\mathbb {Z}}}$, ${{\mathbb {Q}}}$, ${{\mathbb {R}}}$ and ${{\mathbb {C}}}$ respectively. We will write ${\mathbf {t}}= (t_1,\dots , t_d)^T \in {{\mathbb {R}}}^d$ and ${\mathbf{x}}= (x_1,\dots , x_d)^T \in {{\mathbb {R}}}^d$.

Given a (Lebesgue) measurable set $S\subset {{\mathbb {R}^d}}$, $|S|_d$ will denote its Lebesgue measure on ${{\mathbb {R}^d}}$ and $\chi _S$ will be its characteristic function. Given a $d \times d$, $d \ge 1$, matrix M, its transpose will be denoted by $M^T$ and the complex conjugate of its transpose will be denoted by $M^*$. The $d \times d$ identity matrix will be denoted by $\mathbf{I}_{d \times d}$. It is said that a $d \times d$ matrix M is unitary if $M M^* = \mathbf{I}_{d \times d}$.

A linear map $A: {{\mathbb {R}}}^{d} \rightarrow {{\mathbb {R}}}^d$ is said to be expansive if all (complex) eigenvalues of A have modulus greater than 1. The set of all expansive linear maps on ${{\mathbb {R}}}^d$ preserving the lattice ${{\mathbb {Z}}}^d$ such that $A ({{\mathbb {Z}}}^d) \subset {{\mathbb {Z}}}^d$ will be denoted by $\mathbf{E}_d({{\mathbb {Z}}})$. In this paper we emphasize on those linear maps with the additional condition that $|\det A| =2$. This set of linear maps will be denoted by $\mathbf{E}^{(2)}_d({{\mathbb {Z}}})$. Here, with some abuse in the notation, we will denote with the same letter a linear map on ${{\mathbb {R}}}^d$ and its corresponding matrix with respect to the canonical basis of ${{\mathbb {R}}}^d$. Note that if $A \in \mathbf{E}_d({{\mathbb {Z}}})$ then $d_A:= |\det A |$ is an integer greater than 1. The quotient groups ${{\mathbb {Z}}}^{d}/A {{\mathbb {Z}}}^{d}$ and $(A^*)^{-1}{{\mathbb {Z}}}^{d}/{{\mathbb {Z}}}^{d}$ are well defined. By $\varvec{\Delta }_A$ and $\varvec{\Gamma }_A$ we will denote a full collection of representatives of the cosets of ${{\mathbb {Z}}}^{d}/A {{\mathbb {Z}}}^{d}$ and $(A^*)^{-1}{{\mathbb {Z}}}^{d}/ {{\mathbb {Z}}}^{d}$ respectively. Recall that there are exactly $d_A$ cosets [22, 56, p. 109].

Let $A \in \mathbf{E}^{(2)}_d({{\mathbb {Z}}})$. A function $\psi \in L^2({{\mathbb {R}}}^d)$ is called an orthonormal wavelet or simply a wavelet associated to the dilation A, if the system

$$\begin{aligned} \{d_A^{j/2}\psi (A^j \mathbf{x} + \mathbf {k}) ~:~ j \in {{\mathbb {Z}}}, \, \mathbf {k}\in {{\mathbb {Z}^d}}\} \end{aligned}$$

(1)

is an orthonormal basis for $L^2({{\mathbb {R}}}^d)$.

A key tool to work with wavelets is the Fourier transform. We adopt the convention that the Fourier transform of a function $f\in L^1(\mathbb {R}^d) \cap L^2(\mathbb {R}^d)$ is defined by

$$\begin{aligned} {\widehat{f}}(\mathbf{y}) = \int _{\mathbb {R}^n} f (\mathbf{x}) e^{- 2 \pi i \mathbf{x}\cdot \mathbf{y} } \, d\mathbf{x}, \end{aligned}$$

where $\mathbf{x}\cdot \mathbf{y}$ means the inner product in ${{\mathbb {R}}}^d$ of $\mathbf{x}$ and $\mathbf{y}$. The Fourier transform is extended to $L^2({{\mathbb {R}^d}})$ in the usual way.

An orthonormal wavelet $\psi \in L^2({{\mathbb {R}}}^d)$ has vanishing moments of order $m \in \{0,1, \cdots \}$ if ${\widehat{\psi }}$ has a zero of order m at the origin.

A function f will be called symmetric or antisymmetric with respect to a center $\mathbf{C} \in {{\mathbb {R}}}^d$ (see e.g. [41]) if

$$\begin{aligned} f(\mathbf{x})= f ( 2\mathbf{C} - \mathbf{x}), \qquad \text {or} \qquad f(\mathbf{x})= - f ( 2\mathbf{C} - \mathbf{x}) \end{aligned}$$

(2)

respectively. If $f \in L^2({{\mathbb {R}^d}})$, the above equalities can be written as

$$\begin{aligned} {\widehat{f}}(\mathbf{t})= e^{-2 \pi i 2 \mathbf{C} \cdot \mathbf{t} }{\widehat{f}} ( - \mathbf{t}), \qquad \text {or} \qquad {\widehat{f}}(\mathbf{t})= -e^{-2 \pi i 2 \mathbf{C} \cdot \mathbf{t} }{\widehat{f}} ( - \mathbf{t}). \end{aligned}$$

(3)

The remainder of this paper is the following. Under the assumption of the existence of a tile set associated to an $A \in \mathbf{E}_d^{(2)}({{\mathbb {Z}}})$, a construction of symmetric compactly supported scaling functions of a multiresolution analysis appears in Sect. 2. Based on the scaling functions in Sect. 2, a family of antisymmetric compactly supported wavelets with high vanishing moments is constructed in Sect. 3. Sections 4 and 5 are devoted to construct wavelets associated to any $A \in \mathbf{E}_2^{(2)}({{\mathbb {Z}}})$ and to the Quincunx matrix on ${{\mathbb {R}}}^3$ respectively. In Sect. 6, we deal with some special dilation matrices and construct compactly supported wavelets with high vanishing moments and any desired degree of regularity.

2 Construction of Scaling Functions in an MRA

Our construction of compactly supported wavelets is based on multiresolution analysis theory and how a wavelet is obtained from a scaling function. In this section, we give a method for constructing symmetric compactly supported scaling functions in a multiresolution analysis associated to a dilation matrix in $E_d^{(2)}({{\mathbb {Z}}})$.

A multiresolution analysis (MRA) is a general method introduced by Mallat [49] and Meyer [50] for constructing wavelets. Afterwards, a multiresolution analysis with $A \in \mathbf{E}_d({{\mathbb {Z}}})$ was considered (see e.g. [22, 48, 54, 56] or [41]). On ${{{\mathbb {R}}}}^{d}$ by a multiresolution analysis associated to A (A-MRA) we will mean a sequence of closed subspaces $V_j$, $j\in {{\mathbb {Z}}}$ of the Hilbert space $L^2({{\mathbb {R}}}^d)$ that satisfies the following conditions:

• (i) $\forall j\in {{\mathbb {Z}}},\qquad V_j\subset V_{j+1}$;

• (ii) $\forall j\in {{\mathbb {Z}}}, \qquad f(\mathbf{x})\in V_j \Leftrightarrow f(A\mathbf{x}) \in V_{j+1}$;

• (iii) $\overline{\cup _{j\in {{\mathbb {Z}}}} V_j} = L^2({{\mathbb {R}}}^d)$;

• (iv) There exists a function $\phi \in V_0$ such that $\left\{ \phi (\mathbf{x} - \mathbf{k}) ~:~ \mathbf{k}\in {{\mathbb {Z}}}^d \right\}$ is an orthonormal basis for $V_0$. This $\phi$ is usually called a scaling function.

A possible starting point to build a multiresolution analysis is $\phi \in L^2({{\mathbb {R}^d}})$, a function such that $\{\phi ({\mathbf{x}}- {\mathbf {k}}) ~:~ {\mathbf {k}}\in {{\mathbb {Z}^d}}\}$ is an orthonormal system and

$$\begin{aligned} V_j := {\overline{\text{ span }}} \{d_A^{j/2}\phi (A^{j}{\mathbf{x}}- {\mathbf {k}}) ~:~ {\mathbf {k}}\in {{\mathbb {Z}^d}}\}, \quad j\in {{\mathbb {Z}}}, \end{aligned}$$

where the closure is in $L^2({{\mathbb {R}^d}})$. We say that $\phi \in L^2({{\mathbb {R}}}^d)$ generates an A-MRA if the subspaces $V_j$ satisfy condition (i) and (iii) in the definition of A-MRA.

We also need the definition of a “tile” set. A measurable set $S \subset {{\mathbb {R}}}^d$ that satisfies

• (a) $| S \cap (S +{\mathbf {k}}) |_d=0$ for all ${\mathbf {k}} \in {{\mathbb {Z}}}^d{\setminus } \{\mathbf{0} \}$.

• (b) There exists $\{ \mathbf {r}_i \}_{i=0}^{d_A-1}$ a full collection of representatives of the cosets of ${{\mathbb {Z}}}^{d}/ A {{\mathbb {Z}}}^{d}$ such that $A(S)= \cup _{i=0}^{d_A-1}(S + \mathbf {r}_i)$.

• (c) $\cup _{{\mathbf {k}}\in {{\mathbb {Z}}}^d } (S + {\mathbf {k}}) = {{\mathbb {R}}}^d$ except (at most) in a null measurable set.

is usually called a tile.

It is well known, (see e.g. [22, 56]) that if $S \subset {{\mathbb {R}}}^d$ is a measurable set such that $\chi _S$ is a scaling function of an A-MRA, then S is a tile set .

In addition, the set

$$\begin{aligned} S= \left\{ \mathbf {x} \in {{\mathbb {R}}}^d ~:~ \mathbf {x}= \sum _{j=1}^{\infty } A^{-j} \mathbf {b}_j, \quad \mathbf {b}_j \in \Delta _A \right\} \end{aligned}$$

is essentially the only set satisfying (b) in the definition of tile set. Note that for different sets of representatives of the cosets of ${{\mathbb {Z}}}^{d}/ A {{\mathbb {Z}}}^{d}$ one obtains very different tiles sets.

The following proposition provides a method for constructing symmetric compactly supported scaling functions in an A-MRA.

Proposition 1

Let $m \in {{\mathbb {N}}}$. Assume that there exists $S \subset {{\mathbb {R}}}^d$, a tile associated to an $A \in \mathbf{E}_d^{(2)}({{\mathbb {Z}}})$ and $\varvec{\Delta }_A= \{ \mathbf {r}_0, \mathbf {r}_1 \}$, a full collection of representatives of the cosets of ${{\mathbb {Z}}}^{d}/ A {{\mathbb {Z}}}^{d}$ with $\mathbf{r}_0=\mathbf{0}$. Let p be a trigonometric polynomial on ${{\mathbb {R}}}$ such that $p(0)=1$, $|p(t)|^2 + |p(t + {\frac{1}{2}})|^2=1$, $p(t) =0$ if and only if $t \in {\frac{1}{2}}+ {{\mathbb {Z}}}$, and $p(t+ {\frac{1}{2}})= o(|t|^m)$ as $t \rightarrow 0$. In addition, assume that there exists ${c} \in {{\mathbb {Z}}}$ such that $p(-t)= e^{2 \pi i c t} p(t)$. Let

$$\begin{aligned} H(\mathbf {t}):= p (\mathbf {r}_1 \cdot {\mathbf {t}}) \end{aligned}$$

(4)

and define

$$\begin{aligned} {\widehat{\phi }}({\mathbf {t}}):=\prod _{j=1}^{\infty } H((A^*)^{-j}{\mathbf {t}}). \end{aligned}$$

(5)

Then, ${\widehat{\phi }}$ is a well defined non null continuous function in $L^2({{\mathbb {R}}}^d)$ such that $\Vert {\widehat{\phi }} \Vert _{L^2({{\mathbb {R}}}^d)} \le 1$, ${\widehat{\phi }}(\mathbf{0})=1$ and ${\widehat{\phi }}(-{\mathbf {t}})= e^{2 \pi i 2 \mathbf{C} \cdot {\mathbf {t}} }{\widehat{\phi }}({\mathbf {t}})$, where $\mathbf{C}= \frac{c}{2}(A- \mathbf{I}_{d\times d})^{-1}\mathbf {r}_1.$ Moreover, $| {\widehat{\phi }} ({\mathbf {t}}+ {\mathbf {k}})|= o(|{\mathbf {t}}|^m)$ as ${\mathbf {t}} \rightarrow {\mathbf{0}}$ when ${\mathbf {k}} \in {{\mathbb {Z}}}^d{\setminus } \{\mathbf{0}\}$. The function $\phi$ whose Fourier transform ${\widehat{\phi }}$ is defined by (5) has compact support, it is symmetric respect $\mathbf{C}$ and it generates an A-MRA.

Generally speaking, to prove that the system of translated by integers of a refinable function is orthonormal is sometimes a really tough task. For this purpose there are three well-known methods in the literature. Cohen [14] gave the first necessary and sufficient conditions for a trigonometric polynomial H to be a low pass filter of an MRA on $L^2({{\mathbb {R}}})$. One more condition was proved by Lawton [46] in terms of the eigenvalues of a transition operator. The third method essentially consists in checking whether the smoothness exponent $\nu _{2}(H,A)$ is positive or not, see e.g. Han [27] and [33, Theorem 7.2.9]. For the proof of Proposition 1, we will here use the following generalization to the multivariate case of Cohen’s condition, see Bownik [10, Theorem 5].

Theorem A

Let $A \in \mathbf{E}_d({{\mathbb {Z}}})$ and let $\varvec{\Gamma }_A = \{\mathbf {q}_i \}_{i=0}^{d_A-1}$ be a full collection of representatives of the cosets of $(A^*)^{-1}{{\mathbb {Z}}}^{d}/ {{\mathbb {Z}}}^{d}$ with $\mathbf {q}_0=\mathbf {0}$. Let H be a trigonometric polynomial on ${{\mathbb {R}}}^d$ such that $H(\mathbf{0})=1$ and $\sum _{i=0}^{d_A-1} |H({\mathbf {t}} + \mathbf {q}_i)|^2 = 1$. Let $\phi \in L^2({{\mathbb {R}}}^d)$ be the function defined by

$$\begin{aligned} {\widehat{\phi }}({\mathbf {t}}):= \prod _{j=1}^{\infty } H((A^*)^{-j}{\mathbf {t}}) \end{aligned}$$

(6)

The following conditions are equivalent:

(A):

$\{ \phi ({\mathbf{x}}- {\mathbf {k}}) ~:~ {\mathbf {k}} \in {{\mathbb {Z}}}^d \}$ is an orthonormal system.

(B):

There exists a compact set $K \subset {{\mathbb {R}}}^d$ such that

a:

the origin is the interior of K.

b:

$|K|_d=1$ and $|K \cap (K+ {\varvec{\ell }} )|_d=0$, for all ${\varvec{\ell }} \in {{\mathbb {Z}}}^d {\setminus } \{ \mathbf{0} \}$.

c:

$H((A^*)^{-j}{\mathbf {t}}) \ne 0$ for all ${\mathbf {t}} \in K$ and for all $j \in {{\mathbb {N}}}$.

Proof of Proposition 1

Let $\varvec{\Gamma }_A= \{ \mathbf {q}_0, \mathbf {q}_1 \}$ be a full collection of representatives of the cosets of $(A^*)^{-1}{{\mathbb {Z}}}^{d}/ {{\mathbb {Z}}}^{d}$ with $\mathbf{q}_0=\mathbf{0}$. By hypotheses $p(0)=1$, then we have that $H(\mathbf{0})=1$. Further, since $\mathbf {r}_1 \cdot \mathbf {q}_1\in {\frac{1}{2}}+ {{\mathbb {Z}}}$,

$$\begin{aligned} |H(\mathbf{t})|^2 + |H(\mathbf{t} + \mathbf {q}_1)|^2= & {} | p (\mathbf {r}_1 \cdot {\mathbf {t}})|^2 + | p (\mathbf {r}_1 \cdot ( {\mathbf {t}}+ \mathbf {q}_1) )|^2 \nonumber \\= & {} | p (\mathbf {r}_1 \cdot {\mathbf {t}})|^2 + | p \Big (\mathbf {r}_1 \cdot {\mathbf {t}}+ {\frac{1}{2}}\Big )|^2 =1. \end{aligned}$$

(7)

By Theorem 1 in [10], ${\widehat{\phi }}$ is a well defined non null continuous function in $L^2({{\mathbb {R}}}^d)$ with $\Vert {\widehat{\phi }} \Vert _{L^2({{\mathbb {R}}}^d)} \le 1$ and ${\widehat{\phi }}(\mathbf{0})=1$.

That $\phi$ is compactly supported can be proved replicating an argument of [56, p. 79].

We see that $| {\widehat{\phi }} ({\mathbf {t}}+ {\mathbf {k}})|= o(|{\mathbf {t}}|^m)$ as ${\mathbf {t}} \rightarrow \mathbf{0}$ when ${\mathbf {k}} \in {{\mathbb {Z}}}^d{\setminus } \{\mathbf{0}\}$. Fix ${\mathbf {k}} \in {{\mathbb {Z}}}^d{\setminus } \{\mathbf{0}\}$, then there exists $j \in {{\mathbb {N}}}$ such that $(A^*)^{-j}{\mathbf {k}}= \mathbf{k}_1 + \mathbf{q}_1$ for some $\mathbf{k}_1 \in {{\mathbb {Z}}}^d$. We also know that |H| and $| {\widehat{\phi }} |$ are uniformly upper bounded by 1. Then

$$\begin{aligned} | {\widehat{\phi }} ({\mathbf {t}}+ {\mathbf {k}})|= & {} \prod _{\ell = 1}^j | H((A^{*})^{-\ell } {\mathbf {t}} + (A^*)^{-\ell }{\mathbf {k}})| | {\widehat{\phi }} ((A^*)^{-j}({\mathbf {t}}+ {\mathbf {k}}))| \\\le & {} | H((A^{*})^{-j} {\mathbf {t}} + \mathbf{q}_1)| = | p( \mathbf{r}_1 \cdot ((A^{*})^{-j} {\mathbf {t}} + \mathbf{q}_1))| \\= & {} | p\Big ( \mathbf{r}_1 \cdot (A^{*})^{-j} {\mathbf {t}} + {\frac{1}{2}}\Big )|. \end{aligned}$$

By Cauchy–Schwarz inequality and having in mind $(A^*)^{-j}$ as a bounded linear operator,

$$\begin{aligned} \frac{|{\widehat{\phi }} ({\mathbf {t}}+ {\mathbf {k}})|}{\Vert {\mathbf {t}}\Vert ^m}\le & {} \frac{\Big | p\Big ( \mathbf{r}_1 \cdot (A^{*})^{-j} {\mathbf {t}} + {\frac{1}{2}}\Big )\Big |}{|\mathbf{r}_1 \cdot (A^{*})^{-j} {\mathbf {t}} |^m}\frac{|\mathbf{r}_1 \cdot (A^{*})^{-j} {\mathbf {t}} |^m}{\Vert {\mathbf {t}}\Vert ^m}\nonumber \\\le & {} \frac{\Big | p\Big ( \mathbf{r}_1 \cdot (A^{*})^{-j} {\mathbf {t}} + {\frac{1}{2}}\Big )\Big |}{|\mathbf{r}_1 \cdot (A^{*})^{-j} {\mathbf {t}} |^m}\frac{\Vert \mathbf{r}_1\Vert ^m \Vert (A^{*})^{-j}\Vert ^m \Vert {\mathbf {t}} \Vert ^m}{\Vert {\mathbf {t}}\Vert ^m} \longrightarrow 0 \end{aligned}$$

(8)

as ${\mathbf {t}} \rightarrow {\mathbf{0}},$ where the last equality follows by the hypothesis $p(t+ {\frac{1}{2}})= o(|t|^m)$ as $t \rightarrow 0$.

Now, we see the symmetry of $\phi$. We have

$$\begin{aligned} {\widehat{\phi }}(-{\mathbf {t}})= & {} \prod _{j=1}^{\infty } H(-(A^*)^{-j}{\mathbf {t}}) = \prod _{j=1}^{\infty } p(- \mathbf{r}_1 \cdot ((A^*)^{-j}{\mathbf {t}})) \\= & {} \prod _{j=1}^{\infty } e^{2 \pi i c \mathbf{r}_1 \cdot (A^*)^{-j}{\mathbf {t}}} p( \mathbf{r}_1 \cdot ((A^*)^{-j}{\mathbf {t}})) = e^{2 \pi i c (\sum _{j \in {{\mathbb {N}}}} {A^{-j}}{} \mathbf{r}_1) \cdot {\mathbf {t}}} {\widehat{\phi }}({\mathbf {t}})\\= & {} e^{2 \pi i 2 \mathbf{C} \cdot {\mathbf {t}}} {\widehat{\phi }}({\mathbf {t}}), \end{aligned}$$

where $\mathbf{C}= \frac{c}{2} \sum _{j=1}^{\infty } A^{-j} \mathbf {r}_1$. Observe that

$$\begin{aligned} A\mathbf{C}- \mathbf{C}= \frac{c}{2} \sum _{j=0}^{\infty } A^{-j} \mathbf {r}_1 - \frac{c}{2} \sum _{j=1}^{\infty } A^{-j} \mathbf {r}_1 = \frac{c}{2} \mathbf {r}_1, \end{aligned}$$

or equivalently,

$$\begin{aligned} \mathbf{C}= \frac{c}{2}(A- \mathbf{I}_{d\times d})^{-1}\mathbf {r}_1. \end{aligned}$$

Thus, taking the inverse of the Fourier Transform, we have $\phi ({\mathbf{x}})= \phi (2\mathbf{C} - {\mathbf{x}})$. In other words, $\phi$ is symmetric respect $\mathbf{C}$ as we wanted to see.

We see that $\{ \phi ({\mathbf{x}} - {\mathbf {k}}) ~:~{\mathbf {k}} \in {{\mathbb {Z}}}^d \}$ is an orthonormal system. It is enough to prove that H satisfies (B) of Theorem A. Since S is a tile associated to A and $\varvec{\Delta }_A= \{ \mathbf {r}_0, \mathbf {r}_1 \}$, then $\{ \chi _S({\mathbf{x}} - {\mathbf {k}}) ~:~{\mathbf {k}} \in {{\mathbb {Z}}}^d \}$ is an orthonormal system and the refinable equation,

$$\begin{aligned} \widehat{\chi _S} (A^*{\mathbf {t}})= \frac{1}{2} (1 + e^{-2 \pi i \mathbf{r}_1 \cdot {\mathbf {t}}}) \widehat{\chi _S} ({\mathbf {t}}), \end{aligned}$$

holds. Thus, the condition (B) in Theorem A is satisfied for $\frac{1}{2} (1 + e^{-2 \pi i \mathbf{r}_1 \cdot {\mathbf {t}}})$, the low pass filter associated to the scaling function $\chi _S$. Since $p(t)=0$ if and only if $t \in {\frac{1}{2}}+ {{\mathbb {Z}}}$, then the zeros of the both polynomials $\frac{1}{2} (1 + e^{-2 \pi i \mathbf{r}_1 \cdot {\mathbf {t}}})$ and H defined in (4) are the same. This implies that the condition (B) in Theorem A is satisfied also for H.

By definition of ${\widehat{\phi }}$, we have

$$\begin{aligned} {\widehat{\phi }} (A^*{\mathbf {t}})= H({\mathbf {t}}){\widehat{\phi }} ({\mathbf {t}}). \end{aligned}$$

In addition, since ${\widehat{\phi }}$ is continuous with ${\widehat{\phi }}(\mathbf{0})=1$, by Lemma 2 and Theorem 1 in [13], we conclude that $\phi$ generates an A-MRA. This finishes the proof.

3 Construction of Compactly Supported Wavelets

Assuming the existence of a tile set associated to an $A \in \mathbf{E}_d^{(2)}$, we construct a family of antisymmetric compactly supported wavelets with high number of vanishing moments. Our results are based on the univariate complex trigonometric polynomials by Han [30, Theorem 1].

The following result is the main result of this manuscript.

Theorem 1

With the same hypotheses and notation as in Proposition 1, define the function $\psi \in L^2({{\mathbb {R}}}^d)$ by

$$\begin{aligned} {\widehat{\psi }}(A^*\mathbf{t}):= e^{2 \pi i \mathbf{u} \cdot \mathbf{t}} \overline{H({\mathbf {t}} + \mathbf{q}_1)} {\widehat{\phi }}(\mathbf{t}), \end{aligned}$$

(9)

where $\varvec{\Gamma }_A= \{ \mathbf {q}_0, \mathbf {q}_1 \}$ is a full collection of representatives of the cosets of $(A^*)^{-1}{{\mathbb {Z}}}^{d}/ {{\mathbb {Z}}}^{d}$ with $\mathbf{q}_0=\mathbf{0}$ and $\mathbf{u} \in {{\mathbb {Z}}}^d$ such that $\mathbf{q}_1 \cdot \mathbf{u}\in {\frac{1}{2}}+ {{\mathbb {Z}}}$. Then $\psi$ is a compactly supported orthonormal wavelet associated to A with vanishing moments of order m. In addition, if c is even, $\psi$ is symmetric with respect to the point ${\frac{1}{2}}A^{-1} (-2 \mathbf{u} - c \mathbf{r}_1 + 2 \mathbf{C})$, and if c is odd, $\psi$ is antisymmetric with respect to the point ${\frac{1}{2}}A^{-1} (-2 \mathbf{u} - c \mathbf{r}_1 + 2 \mathbf{C})$.

Proof

First, we see that

$\psi$ is a compactly supported orthonormal wavelet associated to A. We denote $H_1(\mathbf{t})= e^{2 \pi i \mathbf{u} \cdot \mathbf{t}} \overline{H({\mathbf {t}} + \mathbf{q}_1)}$. We have

$$\begin{aligned} |H_1(\mathbf{t}) |^2 + |H_1(\mathbf{t} + \mathbf{q}_1) |^2 = |H(\mathbf{t} + \mathbf{q}_1) |^2 + |H(\mathbf{t}+ 2 \mathbf{q}_1) |^2=1, \end{aligned}$$

where the last equality holds because $2 \mathbf{q}_1 \in {{\mathbb {Z}}}^d$ and (7). In addition,

$$\begin{aligned}&H(\mathbf{t}) \overline{H_1(\mathbf{t})} + H(\mathbf{t}+ \mathbf{q}_1) \overline{H_1(\mathbf{t} + \mathbf{q}_1)} \\&\quad = H(\mathbf{t})e^{-2 \pi i \mathbf{u} \cdot \mathbf{t}} H({\mathbf {t}} + \mathbf{q}_1) + H(\mathbf{t}+ \mathbf{q}_1)e^{-2 \pi i \mathbf{u} \cdot (\mathbf{t} + \mathbf{q}_1)} H({\mathbf {t}} ) =0. \end{aligned}$$

Bearing in mind all these computations together Proposition 1 and the procedure to obtain a wavelet from an A-MRA proved in [10, Theorem 4] (see also, e.g. [56, 4] or [41]) we conclude that $\psi$ is an orthonormal wavelet associated to A in $L^2({{\mathbb {R}}}^d)$.

Since $\phi$ has compact support and $H_1$ is a trigonometric polynomial, then that $\psi$ is compactly supported follows.

Now, by the definition of ${\widehat{\psi }}(A^*{\mathbf {t}})$ and repeating the computations in (8), one can see that $\psi$ has vanishing moments of order m.

Finally, we see the symmetry of $\psi$. Assume that $c\in 2{{\mathbb {Z}}}+ 1$. Since $2\mathbf{q}_1 \in {{\mathbb {Z}}}^d$, $\phi$ is symmetric respect to $\mathbf{C}$ and $p(-t)= e^{2 \pi i c t} p(t)$, we have

$$\begin{aligned} {\widehat{\psi }}(-A^*{\mathbf {t}})= & {} e^{-2 \pi i \mathbf{u} \cdot \mathbf{t}} \overline{p( - \mathbf{r}_1 \cdot ( {\mathbf {t}} + \mathbf{q}_1))} e^{2 \pi i 2 \mathbf{C} \cdot {\mathbf {t}}} {\widehat{\phi }}({\mathbf {t}}) \\= & {} e^{-2 \pi i \mathbf{u} \cdot \mathbf{t}} e^{-2 \pi i c ((\mathbf{r}_1 \cdot {\mathbf {t}}) + {\frac{1}{2}})} \overline{ p( \mathbf{r}_1 \cdot ( {\mathbf {t}} + \mathbf{q}_1))} e^{2 \pi i 2 \mathbf{C} \cdot {\mathbf {t}}} {\widehat{\phi }}({\mathbf {t}})\\= & {} -e^{-2 \pi i 2 \mathbf{u} \cdot \mathbf{t}} e^{-2 \pi i c \mathbf{r}_1 \cdot {\mathbf {t}}} e^{2 \pi i 2 \mathbf{C} \cdot {\mathbf {t}}} e^{2 \pi i \mathbf{u} \cdot \mathbf{t}} \overline{H({\mathbf {t}} + \mathbf{q}_1)} {\widehat{\phi }}({\mathbf {t}})\\= & {} -e^{-2 \pi i (2 \mathbf{u} + c \mathbf{r}_1 - 2 \mathbf{C}) \cdot \mathbf{t}} {\widehat{\psi }}(A^*{\mathbf {t}}). \end{aligned}$$

Thus, $\psi$ is antisymmetric with respect to the point ${\frac{1}{2}}A^{-1} (-2 \mathbf{u} -c \mathbf{r}_1 + 2 \mathbf{C})$ as we wanted to see. When $c \in 2{{\mathbb {Z}}}$, that $\psi$ is symmetric with respect to the point ${\frac{1}{2}}A^{-1} (-2 \mathbf{u} -c \mathbf{r}_1 + 2 \mathbf{C})$ follows analogously. This finishes the proof. $\square$

To be more concise in how to apply Theorem 1, we use the following low pass filters by Han [30]. The next theorem is written in a modified form from the original one. It is only adapted to our context.

Theorem B

Let k and n be two positive integers such that $k \le n$. Define

$$\begin{aligned} p_{k,n}(t):= \sum _{j=0}^{k-1} \frac{(2n+2j-3)!!}{(2j)!!(2n-3)!!} t^j, \qquad t \in {{\mathbb {R}}}, \end{aligned}$$

(10)

where $(2j)!!:= (2j)(2j-2) \times \cdots \times 4 \times 2$ and $(2n-3)!!= (2n-3)(2n-5) \times \cdots \times 3 \times 1$. Then

$$\begin{aligned} h_{k,n}(t):= 1- t^{2n-1} [p_{k,n}(1-t) ]^2- (1-t)^{2n-1} [p_{k,n}(t) ]^2 \ge 0, \quad t \in [0,1] \end{aligned}$$

and $t^{k}(1-t)^k | h_{k,n}$, or equivalently,

$$\begin{aligned} h_{k,n}(t) = t^k(1-t)^kR_{k,n}(4t(1-t)) \quad \text {and} \quad R_{k,n}(t) \ge 0, \quad t \in [0,1], \end{aligned}$$

for an unique polynomial $R_{k,n}$ with real coefficients and with $deg(R_{k,n}) \le n-2$.

Define a mask $\widehat{a_{k,n}}$ by

$$\begin{aligned}&\widehat{a_{k,n}}(t):= \widehat{a^r_{k,n}}(t) + i \widehat{a^i_{k,n}}(t), \\&\widehat{a^r_{k,n}}(t):= 2^{1-2n} e^{2 \pi i (n-1) t} ( 1+ e^{-2 \pi i t})^{2n-1} p_{k,n}(\sin ^2 (\pi t)),\\&\widehat{a^i_{k,n}}(t):= 2^{-2k-1} (e^{2 \pi i t} - e^{-2 \pi i t})^k [ q_{k,n}(4\pi t) + (-1)^k e^{-2 \pi i t} q_{k,n}(-4 \pi t)], \end{aligned}$$

where $q_{k,n}$ is a 1-periodic trigonometric polynomial with real coefficients obtained via Riesz lemma from $R_{k,n}$ such that

$$\begin{aligned} |q_{k,n}(t)|^2= R_{k,n}(\sin ^2(\pi t)). \end{aligned}$$

Define $odd_k= (1-(-1)^k)/2$. Then

• (i) $\widehat{a_{k,n}}$ is a symmetric orthogonal mask satisfying $|\widehat{a_{k,n}}(t)|^2 + |\widehat{a_{k,n}}(t + {\frac{1}{2}})|^2=1$ and $\widehat{a_{k,n}}(t) = e^{-2 \pi i t} \widehat{a_{k,n}}(-t)$.

• (ii) $\widehat{a_{k,n}}$ has $k+1-odd_k$ sum rules: $\widehat{a_{k,n}}(t+ {\frac{1}{2}})= O(|t|^{k+1-odd_k})$ as $t \rightarrow 0$.

• (iii) $\widehat{a_{k,n}}$ has $k+odd_k$ linear-phase moments with phase $-1/2$: $\widehat{a_{k,n}}(t)= e^{- \pi i t} + O(|t|^{k+odd_k})$ as $t \rightarrow 0$.

• (iv) $\widehat{a_{k,n}}(t)= 0$ if and only if $t \in {\frac{1}{2}}+ \mathbb {Z}$.

By Proposition 1, Theorem 1 and Theorem B, we have the following.

Corollary 1

Assume that there exists $S \subset {{\mathbb {R}}}^d$, a tile associated to $A \in \mathbf{E}_d^{(2)}({{\mathbb {Z}}})$ and $\varvec{\Delta }_A= \{ \mathbf {r}_0, \mathbf {r}_1 \}$, a full collection of representatives of the cosets of ${{\mathbb {Z}}}^{d}/ A {{\mathbb {Z}}}^{d}$ with $\mathbf{r}_0=\mathbf{0}$.

Let $m, n \in {{\mathbb {N}}}$ such that $m+1\le n$, and $k=m+1$. Let $p(t)=\widehat{a_{k,n}}(t)$ where $\widehat{a_{k,n}}$ is the univariate trigonometric polynomial defined in Theorem B.

Define

$$\begin{aligned} H(\mathbf {t}):= p (\mathbf {r}_1 \cdot {\mathbf {t}}), \end{aligned}$$

and functions $\phi$ and $\psi$ by

$$\begin{aligned} {\widehat{\phi }}({\mathbf {t}}):=\prod _{j=1}^{\infty } H((A^*)^{-j}{\mathbf {t}}) \quad \text {and} \quad {\widehat{\psi }}(A^*\mathbf{t}):= e^{2 \pi i \mathbf{r}_1 \cdot \mathbf{t}} \overline{H({\mathbf {t}} + \mathbf{q}_1)} {\widehat{\phi }}(\mathbf{t}) \end{aligned}$$

respectively, where $\varvec{\Gamma }_A= \{ \mathbf {q}_0, \mathbf {q}_1 \}$ is a full collection of representatives of the cosets of $(A^*)^{-1}{{\mathbb {Z}}}^{d}/ {{\mathbb {Z}}}^{d}$ with $\mathbf{q}_0=\mathbf{0}$. Then,

• (i) The function ${\widehat{\phi }}$ is a well defined non null continuous function in $L^2({{\mathbb {R}}}^d)$ such that $\Vert {\widehat{\phi }} \Vert _{L^2({{\mathbb {R}}}^d)} \le 1$, ${\widehat{\phi }}(\mathbf{0})=1$ and ${\widehat{\phi }}(-{\mathbf {t}})= e^{2 \pi i 2 \mathbf{C} \cdot {\mathbf {t}} }{\widehat{\phi }}({\mathbf {t}})$ where $\mathbf{C}= \frac{1}{2}(A- \mathbf{I}_{d\times d})^{-1}\mathbf {r}_1.$

  Moreover, $| {\widehat{\phi }} ({\mathbf {t}}+ {\mathbf {k}})|= o(|{\mathbf {t}}|^m)$ as ${\mathbf {t}} \rightarrow {\mathbf{0}}$ when ${\mathbf {k}} \in {{\mathbb {Z}}}^d{\setminus } \{\mathbf{0}\}$.

• (ii) The function $\phi$ has compact support, it is symmetric respect to $\mathbf{C}$ and it generates an A-MRA.

• (iii) The function $\psi$ is a compactly supported orthonormal wavelet associated to A. In addition, $\psi$ has vanishing moments of order m and it is antisymmetric with respect to the point ${\frac{1}{2}}A^{-1} ( 2 \mathbf{C} - 3\mathbf{r}_1)$.

According to [42, Theorem 5.1] (see Theorem D below) and [11, Proposition 2], it is known that there exists tiles sets associated to any $A \in \mathbf{E}_d^{(2)}({{\mathbb {Z}}})$ with $d=2,3$. Then, by Corollary 1 we have the following.

Corollary 2

Let $m \in {{\mathbb {N}}}$. Let $A \in \mathbf{E}_d^{(2)}({{\mathbb {Z}}})$ with $d=2,3$. There exists $\psi \in L^2({{\mathbb {R}}}^d)$ a compactly supported wavelet associated to A with vanishing moments of order m and it is antisymmetric with respect some point on ${{\mathbb {R}}}^d$.

4 Some Bivariate Wavelets

Here, we focus on compactly supported wavelets defined in $L^2({{\mathbb {R}}}^2)$. Indeed, for a fixed $A \in \mathbf{E}_2^{(2)}({{\mathbb {Z}}})$ we remove the hypothesis of the existence of a tile set associated to A in Corollary 1. Based on the proof of Theorem 5.1 by Lagarias and Wang [42], we construct antisymmetric compactly supported wavelets with any desired number of vanishing moments associated to any fix $A \in \mathbf{E}_2^{(2)}({{\mathbb {Z}}})$. In this section, $\mathbf {t}=(t_1,t_2) \in \mathbb {R}^2$.

Let us make the following remarks on all expansive linear maps on ${{\mathbb {R}}}^2$ that preserve the integer lattice.

Two matrices A and B with integer coefficients are integrally similar if there exists a matrix U with integer entries such that $|\det U| =1$ and $A= U^{-1} B U$. Let

$$\begin{aligned} A_1:= \left( \begin{array}{rrrrrrrrrrrrrrrr} 0 &{} 2 \\ 1 &{} 0 \end{array} \right) , \ A_2:= \left( \begin{array}{rrrrrrrrrrrrrrrr} 0 &{} 2 \\ -1 &{} 0 \end{array} \right) , \ A_3:= \left( \begin{array}{rrrrrrrrrrrrrrrr} 0 &{} 2 \\ -1 &{} 1 \end{array} \right) , \nonumber \\ \ A_4:= \left( \begin{array}{rrrrrrrrrrrrrrrr} 0 &{} -2 \\ 1 &{} -1 \end{array} \right) , \ A_5:= \left( \begin{array}{rrrrrrrrrrrrrrrr} 1 &{} 1 \\ -1 &{} 1 \end{array} \right) , \ A_6:= \left( \begin{array}{rrrrrrrrrrrrrrrr} -1 &{} -1 \\ 1 &{} -1 \end{array} \right) . \end{aligned}$$

(11)

The following complete classification of all matrices in $\mathbf{E}^{(2)}_2({{\mathbb {Z}}})$ was proved in [42, Lemma 5.2 ].

Lemma C

Let $A \in \mathbf{E}_2({{\mathbb {Z}}})$. If $\det A = -2$ then A is integrally similar to $A_1$. If $\det A = 2$ then A is integrally similar to one of the matrices $A_{\ell }$, $\ell = 2,\dots , 6$.

Observe that the set

$$\begin{aligned} \Lambda = \{ \mathbf {r}_0= (0,0)^T, \mathbf {r}_1= (1,0)^T \}, \end{aligned}$$

(12)

is a full collection of representatives of the cosets of ${{\mathbb {Z}}}^{2}/A_{\ell } {{\mathbb {Z}}}^{2}$, $\ell = 1,\dots , 6$.

The proof of Theorem 5.1 in [42] really says the following.

Theorem D

Let $A \in \mathbf{E}_2^{(2)}({{\mathbb {Z}}})$ and let $\ell \in \{ 1,\dots , 6 \}$ be such that there exists a matrix U with integer entries such that $|\det U| =1$ and $A= U^{-1} A_{\ell } U$. Define

$$\begin{aligned} S= \{ \mathbf {x} \in {{\mathbb {R}}}^n ~:~ \mathbf {x}= \sum _{j=1}^{\infty } A^{-j} \mathbf {b}_j, \quad \mathbf {b}_j \in \Delta _A \}, \end{aligned}$$

where $\Delta _A= \{ (0,0)^T, U^{-1}(1,0)^T \}$. Then, the function $\chi _S$ generates an A-MRA in $L^2({{\mathbb {R}}}^2)$.

As a consequence of Theorem D and of the construction made in Corollary 1, we have the following wavelets.

Corollary 3

Let $A \in \mathbf{E}_2^{(2)}({{\mathbb {Z}}})$ and let $\ell \in \{ 1,\dots , 6 \}$ be such that there exists a matrix U with integer entries such that $|\det U| =1$ and $A= U^{-1} A_{\ell } U$. Let $\mathbf{r}_1= U^{-1}(1,0)^T$. Let $\varvec{\Gamma }_A= \{ \mathbf {q}_0, \mathbf {q}_1 \}$ be a full collection of representatives of the cosets of $(A^*)^{-1}{{\mathbb {Z}}}^{2}/ {{\mathbb {Z}}}^{2}$ with $\mathbf{q}_0=\mathbf{0}$.

Let $m, n \in {{\mathbb {N}}}$ such that $m < n$, and $k=m+1$. Let $p(t)=\widehat{a_{k,n}}(t)$ where $\widehat{a_{k,n}}$ is the univariate trigonometric polynomial defined in Theorem B.

Define

$$\begin{aligned} H(\mathbf {t}):= p (\mathbf {r}_1 \cdot {\mathbf {t}}), \end{aligned}$$

and functions $\phi$ and $\psi$ by

$$\begin{aligned} {\widehat{\phi }}({\mathbf {t}}):=\prod _{j=1}^{\infty } H((A^*)^{-j}{\mathbf {t}}) \quad \text {and} \quad {\widehat{\psi }}(A^*\mathbf{t}):= e^{2 \pi i \mathbf{r}_1 \cdot \mathbf{t}} \overline{H({\mathbf {t}} + \mathbf{q}_1)} {\widehat{\phi }}(\mathbf{t}). \end{aligned}$$

Then,

• (i) The function $\phi$ has compact support, it is symmetric respect to $\mathbf{C}= \frac{1}{2}(A- \mathbf{I}_{2\times 2})^{-1}\mathbf {r}_1$ and it generates an A-MRA.

• (ii) The function $\psi$ is a compactly supported orthonormal wavelet associated to A. In addition, $\psi$ has vanishing moments of order m and it is antisymmetric with respect to the point ${\frac{1}{2}}A^{-1} ( 2 \mathbf{C} - 3\mathbf{r}_1)$.

We now focus on the particular case where a matrix is integrally similar to $A_1$. We have the following tiles.

Proposition 2

Let ${\varepsilon }_1, {\varepsilon }_2= \pm 1$. Let $A \in \mathbf{E}_2^{(2)}({{\mathbb {Z}}})$ and let U be an $2 \times 2$ matrix with integer entries such that $|\det U| =1$ and $A= U^{-1} A_1 U$. Define

$$\begin{aligned} S= \{ \mathbf {x} \in {{\mathbb {R}}}^2 ~:~ \mathbf {x}= \sum _{j=1}^{\infty } A^{-j} \mathbf {b}_j, \quad \mathbf {b}_j \in \Delta _A \}, \end{aligned}$$

(13)

where $\Delta _A= \{ \mathbf{r}_0= (0,0)^T, \mathbf{r}_1= U^{-1}({\varepsilon }_1 , {\varepsilon }_2 )^T \}$. Then, the function $\chi _S$ generates an A-MRA in $L^2({{\mathbb {R}}}^2)$.

Proof

First, we prove the statement for $A=A_1$ and ${\varepsilon }_1= {\varepsilon }_2= \pm 1$.

It is not hard to see that $\{ (0,0), ({\varepsilon }_1 , {\varepsilon }_1 ) \}$ is a full collection of representatives of the cosets of ${{\mathbb {Z}}}^{2}/ A_1{{\mathbb {Z}}}^{2}$. Define

$$\begin{aligned} S_1= \{ \mathbf {x} \in {{\mathbb {R}}}^2 ~:~ \mathbf {x}= \sum _{j=1}^{\infty } A_1^{-j} \mathbf {b}_j, \quad \mathbf {b}_j \in \Delta _{A_1} \}, \end{aligned}$$

(14)

where $\Delta _{A_1}= \{ \mathbf{r}_0= (0,0)^T, \mathbf{r}_1= ({\varepsilon }_1 , {\varepsilon }_1)^T \}$. By definition of $S_1$, we have $A_1S_1= S_1 \cup (S_1+ \mathbf{r}_1)$. Thus $\widehat{\chi _{S_1}} (\mathbf{0}) \ne 0$ and

$$\begin{aligned} \widehat{\chi _{S_1}} (A_1^*{\mathbf {t}})= \frac{1}{2} (1 + e^{- 2 \pi i {\varepsilon }_1 (t_1 + t_2) }) \widehat{\chi _{S_1}} ({\mathbf {t}})= H_1({\mathbf {t}}) \widehat{\chi _{S_1}} ({\mathbf {t}}), \end{aligned}$$

where $H_1({\mathbf {t}})= \frac{1}{2} (1 + e^{-2 \pi i {\varepsilon }_1 (t_1 + t_2) })$.

To check that $\{ \chi _{S_1}({\mathbf{x}} - {\mathbf {k}}) : {\mathbf {k}} \in {{\mathbb {Z}}}^2)\}$ is an orthonormal system, we see that H satisfies the condition (B) in Theorem A. By hypotheses, we have that $H_1({\mathbf {t}})=0$ if and only if ${\mathbf {t}}= (t_1,t_2) \in Z$ where

$$\begin{aligned} Z= \left\{ (t_1,t_2) \in {{\mathbb {R}}}^2 ~:~ t_1 + t_2 = n + {\frac{1}{2}}, \, n \in {{\mathbb {Z}}}\right\} . \end{aligned}$$

By induction,

$$\begin{aligned} (A_1^*)^{2\ell } = 2^{\ell }\left( \begin{array}{ccccc} 1 &{} 0 \\ 0&{} 1 \end{array} \right) \quad \text {and} \quad (A_1^*)^{2\ell -1} = 2^{\ell -1}\left( \begin{array}{ccccc} 0 &{} 1 \\ 2 &{} 0 \end{array} \right) , \quad \ell \in {{\mathbb {N}}}. \end{aligned}$$

Then, it is not hard to see that $(A^*)^jZ \subset R$ where

$$\begin{aligned} R= \{ (t_1, -t_1 + k) \in {{\mathbb {R}}}^2 ~:~ k \in {{\mathbb {Z}}}{\setminus } \{ 0 \} \} \cup \{ (t_1, -2t_1 + k) \in {{\mathbb {R}}}^2 ~:~ k \in {{\mathbb {Z}}}{\setminus } \{0 \} \}. \end{aligned}$$

We take $K_1 \subset {{\mathbb {R}}}^2$ the parallelogram with vertices $\big (- \frac{1}{2}, \frac{11}{10}\big )$, $\big ( \frac{1}{2}, - \frac{1}{10}\big )$, $\big ( \frac{1}{2}, -\frac{11}{10}\big )$ and $\big ( -\frac{1}{2}, \frac{1}{10}\big )$ (Fig. 1).

Fig. 1

The sets $K_1$ and R associated to $H_1({\mathbf {t}})$ and $A_1$

It is easy to see that the origin is in the interior of $K_1$ and $K_1 + {{\mathbb {Z}}}^2 = {{\mathbb {R}}}^2$. Moreover, since $\frac{11}{10} < \frac{1}{2} + 1$ and $- \frac{1}{10} < 0= - 2 \frac{1}{2} +1$, we have that $K_1$ is in the region bounded by the lines $t_2= -t_1 + 1$, $t_2= -2t_1 + 1$, $t_2= -t_1 -1$ and $t_2= -2t_1 - 1$. Hence $K_1 \bigcap (A_1^*)^jZ = \emptyset$, $j \in {{\mathbb {N}}}$, or equivalently, $(A_1^*)^{-j}K_1 \bigcap Z = \emptyset$, $j \in {{\mathbb {N}}}$. Thus, $K_1$ satisfies the condition (B) in Theorem A. This, together Lemma 2 and Theorem 1 in [13] imply that $\chi _{S_1}$ is a scaling function of an $A_1$-MRA.

For the general case $A=U^{-1}A_1 U$, having in mind that $\{ (0,0), U^{-1}({\varepsilon }_1 , {\varepsilon }_1 ) \}$ is a full collection of representatives of the cosets of ${{\mathbb {Z}}}^{2}/ A{{\mathbb {Z}}}^{2}$ follows, U is an integer matrix and $|\det U|$=1, we get that $U^{*}K_1$ satisfies the condition (B) in Theorem A with respect to A.

When $\varepsilon _1=-\varepsilon _2 = \pm 1$, the proof is analogous if we take $K_1 \subset {{\mathbb {R}}}^2$, the parallelogram with vertices $\big ( \frac{1}{2}, -\frac{11}{10}\big )$, $\big ( -\frac{1}{2}, \frac{1}{10}\big )$, $\big (- \frac{1}{2}, \frac{11}{10}\big )$ and $\big ( \frac{1}{2}, - \frac{1}{10}\big )$. $\square$

By Proposition 2 and Corollary 1, we have the following wavelets.

Corollary 4

Let $A \in \mathbf{E}_2^{(2)}({{\mathbb {Z}}})$ such that $A= U^{-1} A_1 U$ where U is a matrix with integer entries such that $|\det U| =1$.

Let $\mathbf{r}_1= U^{-1}({\varepsilon }_1,{\varepsilon }_2)^T$ where ${\varepsilon }_i= \pm 1$, $i=1,2$. Let $\varvec{\Gamma }_A= \{ \mathbf {q}_0, \mathbf {q}_1 \}$ be a full collection of representatives of the cosets of $(A^*)^{-1}{{\mathbb {Z}}}^{2}/ {{\mathbb {Z}}}^{2}$ with $\mathbf{q}_0=\mathbf{0}$.

Let $m, n \in {{\mathbb {N}}}$ such that $m < n$, and $k=m+1$. Let $p(t)=\widehat{a_{k,n}}(t)$ where $\widehat{a_{k,n}}$ is the univariate trigonometric polynomial defined in Theorem B.

Define

$$\begin{aligned} H(\mathbf {t}):= p (\mathbf {r}_1 \cdot {\mathbf {t}}), \end{aligned}$$

and functions $\phi$ and $\psi$ by

$$\begin{aligned} {\widehat{\phi }}({\mathbf {t}}):=\prod _{j=1}^{\infty } H((A^*)^{-j}{\mathbf {t}}) \quad \text {and} \quad {\widehat{\psi }}(A^*\mathbf{t}):= e^{2 \pi i \mathbf{r}_1 \cdot \mathbf{t}} \overline{H({\mathbf {t}} + \mathbf{q}_1)} {\widehat{\phi }}(\mathbf{t}). \end{aligned}$$

Then,

• (i) The function $\phi$ has compact support, it is symmetric respect to $\mathbf{C}= \frac{1}{2}(A- \mathbf{I}_{2\times 2})^{-1}\mathbf {r}_1$ and it generates an A-MRA.

• (ii) The function $\psi$ is a compactly supported orthonormal wavelet associated to A. In addition, $\psi$ has vanishing moments of order m and it is antisymmetric with respect to the point $\frac{1}{2}A^{-1}(2\mathbf{C} - 3 \mathbf{r}_1)$.

Below, the reader can find some concrete examples of H, trigonometric polynomials used in Corollaries 3 and 4.

Furthermore of those corollaries, we give estimations for $\nu _{2}(H,A)$, the smoothness exponent of H and a dilation A. To do so, we will require Theorem 7.2.9 of [33], although it is also possible to be calculated by [27, Algorithm 2] using the symmetry of H.

Example 1

Let $k=n=2$. We obtain the next univariate low pass filter

$$\begin{aligned} \widehat{a^{r}_{2,2}}(t):= & {} -\frac{3}{64}(e^{-2\pi i t})^{-2} + \frac{5}{64}(e^{-2\pi i t})^{-1} + \frac{15}{32} + \frac{15}{32}e^{-2\pi i t} \\&+ \frac{5}{64}(e^{-2\pi i t})^{2} - \frac{3}{64}(e^{-2\pi i t})^3 \end{aligned}$$

and

$$\begin{aligned} \widehat{a^{i}_{2,2}}(t):= & {} \frac{\sqrt{15}}{64}(e^{-2\pi i t})^{-2} + \frac{\sqrt{15}}{64}(e^{-2\pi i t})^{-1} - \frac{\sqrt{15}}{32} - \frac{\sqrt{15}}{32}e^{-2\pi i t} \\&+ \frac{\sqrt{15}}{64}(e^{-2\pi i t})^{2} + \frac{\sqrt{15}}{64}(e^{-2\pi i t})^3. \end{aligned}$$

So, we have

$$\begin{aligned} \widehat{a_{2,2}}(t):=\widehat{a_{2,2}^{r}}(t)+ i\widehat{a_{2,2}^{i}}(t) \end{aligned}$$

and our multivariate trigonometric polynomial in Corollary 3 and 4 is

$$\begin{aligned} H(\mathbf {t}):= \widehat{a_{2,2}}(\mathbf {r}_{1} \cdot \mathbf {t}). \end{aligned}$$

By using Theorem 7.2.9 of [33], we compute the smoothness exponent in dimension 2 with the dilation matrices $A_{\ell }, \ell =1, \ldots , 6$ in (11). The following table summarizes our computations (Table 1).

Table 1 The smoothness exponents associated to the multivariate polynomial $\widehat{a_{2,2}}$ and dilation matrices $A_{\ell }$, $\ell =1, \ldots , 6$

Following the same structure as the previous example, we have:

Example 2

When $k=n=3$, we consider the trigonometric polynomial given by Example 1 in [30]. Let

$$\begin{aligned} \widehat{a^{r}_{3,3}}(t):= & {} \frac{35}{4096}(e^{-2\pi i t})^{-4} - \frac{45}{4096}(e^{-2\pi i t})^{-3} - \frac{63}{1024}(e^{-2\pi i t})^{-2} + \frac{105}{1024}(e^{-2\pi i t})^{-1} \\&+ \frac{945}{2048} + \frac{945}{2048}e^{-2\pi i t}+ \frac{105}{1024}(e^{-2\pi i t})^{2} - \frac{63}{1024}(e^{-2\pi i t})^3 - \frac{45}{4096}(e^{-2\pi i t})^4 \\&+ \frac{35}{4096}(e^{-2\pi i t})^5 , \\ \widehat{a^{i}_{3,3}}(t):= & {} \left( \frac{\sqrt{210}}{32\sqrt{94}-256} \right) \left( -\frac{15}{128}(e^{-2\pi i t})^{-4} + \left( -\frac{79}{128} + \frac{\sqrt{94}}{16}\right) (e^{-2\pi i t})^{-3} \right. \\&+ \left( \frac{31}{32} - \frac{\sqrt{94}}{16} \right) (e^{-2\pi i t})^{-2} + \left( \frac{63}{32} - \frac{3\sqrt{94}}{16}\right) (e^{-2\pi i t})^{-1} \\&-\frac{141}{64} + \frac{3\sqrt{94}}{16} + \left( -\frac{141}{64} + \frac{3\sqrt{94}}{16}\right) e^{-2\pi i t} + \left( \frac{63}{32} - \frac{3\sqrt{94}}{16} \right) (e^{-2\pi i t})^{2} \\&+ \left( \frac{31}{32} - \frac{\sqrt{94}}{16} \right) (e^{-2\pi i t})^{3} + \left( -\frac{79}{128} + \frac{\sqrt{94}}{16}\right) (e^{-2\pi i t})^{4} \\&-\frac{15}{128}(e^{-2\pi i t})^{5} \left. \right) , \end{aligned}$$

and then

$$\begin{aligned} \widehat{a_{3,3}}(t):=\widehat{a_{3,3}^{r}}(t)+ i\widehat{a_{3,3}^{i}}(t). \end{aligned}$$

So, our multivariate trigonometric polynomial is

$$\begin{aligned} H(\mathbf {t}):= \widehat{a_{3,3}}(\mathbf {r}_{1} \cdot \mathbf {t}). \end{aligned}$$

Thus, the smoothness exponent in dimension 2 with the dilation matrices $A_{\ell }$, $\ell =1, \ldots , 6$ in (11) appears in Table 2.

Table 2 The smoothness exponents associated to the multivariate polynomial $\widehat{a_{3,3}}$ and dilation matrices $A_{\ell }$, $\ell =1, \ldots , 6$

We want to finish this section by showing two more examples in numerical form.

Example 3

Let $k=2$ and $n=3$ in Theorem B. Let

$$\begin{aligned} \widehat{a^{r}_{2,3}}(t):= & {} -\,0.019531250(e^{-2\pi i t})^{-3} -0.027343750(e^{-2 \pi i t})^{-2} \\&+\,0.13671875(e^{-2\pi i t})^{-1} + 0.41015625 + 0.41015625e^{-2\pi i t} \\&+\,0.13671875(e^{-2\pi i t})^2 -0.027343750(e^{-2\pi i t})^3 \\&-\,0.019531250(e^{-2\pi i t})^4 \end{aligned}$$

and

$$\begin{aligned} \widehat{a^{i}_{2,3}}(t):= & {} -\,0.09789420359(e^{-2\pi i t})^{-3} +0.005455456988(e^{-2\pi i t})^{-2} \\&+\,0.2012438642(e^{-2\pi i t})^{-1} -0.1088051176 -0.1088051176(e^{-2\pi i t}) \\&+\, 0.2012438642(e^{-2\pi i t})^2 + 0.005455456988(e^{-2\pi i t})^3 \\&-\,0.09789420359(e^{-2\pi i t})^4 \end{aligned}$$

then we have

$$\begin{aligned} \widehat{a_{2,3}}(t):=\widehat{a_{2,3}^{r}}(t)+ i\widehat{a_{2,3}^{i}}(t) \end{aligned}$$

and

$$\begin{aligned} H(\mathbf {t}):= \widehat{a_{2,3}}(\mathbf {r}_{1} \cdot \mathbf {t}). \end{aligned}$$

The smoothness exponent of H with the dilation matrices $A_{\ell }, \ell =1, \ldots , 6$ in (11) appears in the Table 3 below.

Table 3 The smoothness exponents associated to the multivariate polynomial $\widehat{a_{2,3}}$ and dilation matrices $A_{\ell }$, $\ell =1, \ldots , 6$

Example 4

Let $k=n=4$ in Theorem B, we work with the trigonometric polynomial given by Example 2 in [30]. Let

$$\begin{aligned} \widehat{a^{r}_{4,4}}(t):= & {} -\,0.0017623901367(e^{-2 \pi i t})^{-6} +0.0020828247070(e^{-2\pi i t})^{-5} \\&+\,0.015274047852(e^{-2\pi i t})^{-4} -0.019638061523(e^{-2\pi i t})^{-3} \\&-\,0.068733215332(e^{-2\pi it})^{-2} + 0.11455535889(e^{-2\pi i t})^{-1} \\&+\, 0.45822143555 + 0.45822143555e^{-2\pi i t} + 0.11455535889(e^{-2\pi i t})^{2} \\&-\,0.068733215332(e^{-2\pi i t})^{3} -0.019638061523(e^{-2\pi i t})^{4} \\&+\, 0.015274047852(e^{-2\pi i t})^{5} + 0.0020828247070(e^{-2\pi i t})^{6} \\&-\,0.0017623901367(e^{-2\pi i t})^{7} \end{aligned}$$

and

$$\begin{aligned} \widehat{a^{i}_{4,4}}(t):= & {} 0.01549716987(e^{-2\pi i t})^{-6} + 0.0002368658117(e^{-2\pi i t})^{-5} \\&-\,0.06434389801(e^{-2\pi i t})^{-4} -0.003302681777(e^{-2\pi i t})^{-3} \\&+\, 0.1026407591(e^{-2\pi i t})^{-2} + 0.02633923885(e^{-2\pi i t})^{-1} \\&-\,0.07706745391 -0.07706745391(e^{-2\pi i t}) + 0.02633923885(e^{-2\pi i t})^2 \\&+\,0.1026407591(e^{-2\pi i t})^3 -0.003302681777(e^{-2\pi i t})^4 \\&-\,0.06434389801(e^{-2\pi i t})^5 + 0.0002368658117(e^{-2\pi i t})^6 \\&+\,0.01549716987(e^{-2\pi i t})^7 \end{aligned}$$

to obtain the trigonometric polynomials

$$\begin{aligned} \widehat{a_{4,4}}(t):=\widehat{a_{4,4}^{r}}(t)+ i\widehat{a_{4,4}^{i}}(t) \end{aligned}$$

and

$$\begin{aligned} H(\mathbf {t}):= \widehat{a_{4,4}}(\mathbf {r}_{1} \cdot \mathbf {t}). \end{aligned}$$

Then, we can see the smoothness exponent in dimension 2 with the dilation matrices $A_{\ell }, \ell =1, \ldots , 6$ in (11) in the next Table 4.

Table 4 The smoothness exponents associated to the multivariate polynomial $\widehat{a_{4,4}}$ and dilation matrices $A_{\ell }$, $\ell =1, \ldots , 6$

5 Wavelets with the Quincunx Dilation on ${{\mathbb {R}}}^3$

Under the hypothesis of the existence of a compactly supported scaling function, Lan, Zhengxing and Yongdong [44] provided a method for compactly supported constructing Riesz wavelet basis in $L^2({{\mathbb {R}}}^3)$ with symmetry and vanishing moments associated to the $3 \times 3$ Quincunx matrix

$$\begin{aligned} Q= \left( \begin{array}{ccc} 1 &{} 0 &{} 1\\ -1 &{} -1 &{} 1\\ 0 &{} -1 &{} 0 \end{array} \right) . \end{aligned}$$

In the paper by Karoui [36], it is said that the construction made for dyadic dilations provides candidates to be wavelets with Q. In this section, we give a method for constructing antisymmetric compactly supported wavelets with high number of vanishing moments associated to Q.

It is not hard to see that $\Delta _Q^{(1)}= \{ \mathbf{r}_0^{(1)}= (0,0,0)^T, \mathbf{r}_1^{(1)}= (1, 0,0 )^T \}$ and $\Delta _Q^{(2)}= \{ \mathbf{r}_0^{(2)}= (0,0,0)^T, \mathbf{r}_1^{(2)}= (0,1,0 )^T \}$ are full collections of representatives of the cosets of ${{\mathbb {Z}}}^3/Q {{\mathbb {Z}}}^3$. In this section, ${\mathbf {t}}=(t_1,t_2,t_3)^T\in {{\mathbb {R}}}^3$.

We have the following tiles.

Proposition 3

Let $a \in \{1,2\}$, then the function $\chi _{S^{(a)}}$, where

$$\begin{aligned} S^{(a)}= \{ \mathbf {x} \in {{\mathbb {R}}}^3 ~:~ \mathbf {x}= \sum _{j=1}^{\infty } Q^{-j} \mathbf {b}_j, \quad \mathbf {b}_j^{(a)} \in \Delta _Q^{(a)} \}, \end{aligned}$$

(15)

generates an Q-MRA in $L^2({{\mathbb {R}}}^3)$.

Proof

We prove explicitly the statement when $a=1$. When $a=2$ the proof is similar. First, we see that the trigonometric polynomial $H({\mathbf {t}})= \frac{1}{2}(1 + e^{-2 \pi i t_1})$ satisfies the condition (B) in Theorem A. We see that $H({\mathbf {t}})=0$ if and only if ${\mathbf {t}}\in Z$ where

$$\begin{aligned} Z=\{ \mathbf {t}\in {{\mathbb {R}}}^3 \, : \, t_1 = \frac{2n+1}{2}, \, n \in {{\mathbb {Z}}}\}. \end{aligned}$$

For $m \in {{\mathbb {N}}}\cup \{ 0 \}$,

• $(Q^*)^{(3m+1)}(Z)= \{(t_1,t_2,t_3) \in {{\mathbb {R}}}^3 \, : \, t_1+t_3=2^{m}(2n+1), \, n \in {{\mathbb {Z}}}\}$,

• $(Q^*)^{(3m+2)}(Z)= \{ (t_1,t_2,t_3) \in {{\mathbb {R}}}^3 \, : \, t_1-t_2=2^{m}(2n+1) \, n \in {{\mathbb {Z}}}\}$,

• $( Q^*)^{(3m+3)}(Z)= \{ (t_1,t_2,t_3) \in {{\mathbb {R}}}^3 \, : \, t_1=2^{m}(2n+1) \, n \in {{\mathbb {Z}}}\}.$

Observe that

$$\begin{aligned} (Q^*)^{j}(Z) \subset R, \quad j \in {{\mathbb {N}}}, \end{aligned}$$

where

$$\begin{aligned} R= & {} \{ (t_1,t_2,t_3) \in {{\mathbb {R}}}^3 \, : \, t_1+t_3 = k, \quad k \in \mathbb {Z} {\setminus } \{0 \} \} \\&\cup \{(t_1,t_2,t_3) \in {{\mathbb {R}}}^3 \, : \, t_1-t_2 = k, \quad k \in \mathbb {Z} {\setminus } \{0 \} \} \\&\, \, \, \cup \{(t_1,t_2,t_3) \in {{\mathbb {R}}}^3 \, : \, t_1 = k, \quad k \in \mathbb {Z} {\setminus } \{0 \} \}. \end{aligned}$$

On the other hand, we need the following sets:

$$\begin{aligned} L_1= & {} \left[ \frac{7}{16}, \frac{1}{2}\right] \times \left[ -\frac{1}{2}, -\frac{7}{16}\right] \times \left[ -\frac{1}{2},\frac{1}{4}\right] ,\\ L_2= & {} \left[ \frac{7}{16}, \frac{1}{2}\right] \times \left[ -\frac{1}{2}, 0\right] \times \left[ \frac{1}{4},\frac{1}{2}\right] , \\ L_3= & {} \left[ \frac{7}{16}, \frac{1}{2}\right] \times \left[ 0,\frac{1}{2}\right] \times \left[ \frac{1}{4},\frac{1}{2}\right] , \\ L_4= & {} \left[ -\frac{1}{2}, -\frac{7}{16}\right] \times \left[ \frac{7}{16},\frac{1}{2}\right] \times \left[ -\frac{1}{4},\frac{1}{2}\right] , \\ L_5= & {} \left[ -\frac{1}{2}, -\frac{7}{16}\right] \times \left[ 0,\frac{1}{2}\right] \times \left[ -\frac{1}{2},-\frac{1}{4}\right] , \\ L_6= & {} \left[ -\frac{1}{2}, -\frac{7}{16}\right] \times \left[ -\frac{1}{2},0\right] \times \left[ -\frac{1}{2},-\frac{1}{4}\right] , \end{aligned}$$

and

$$\begin{aligned} K= & {} \overline{\left[ -\frac{1}{2}, \frac{1}{2}\right] ^3 {\setminus } (\cup _{\ell =1}^6 L_{\ell }) }\nonumber \\&\cup (L_{1} + (0,1,0)) \cup (L_{2} + (-1,0,0)) \cup (L_{3} + (0,0,-1)) \nonumber \\&\, \, \, \cup (L_{4} + (0,-1,0)) \cup (L_{5} + (1,0,0)) \cup (L_{6} + (0,0,1)). \end{aligned}$$

(16)

Now, we see that this set K satisfies the condition (B) in Theorem A where the dilation is given by the quincunx matrix Q and $H({\mathbf {t}})= \frac{1}{2}(1 + e^{-2 \pi i t_1})$.

By construction, K is a compact set in ${{\mathbb {R}}}^3$, the origin is the interior of K, $|K|_3=1$ and $|K \cap (K+ \ell )|_3=0,$ for all $\ell \in {{\mathbb {Z}}}^3 {\setminus } \{\mathbf{0}\}$.

Since

$$\begin{aligned} \left[ -\frac{1}{2},\frac{1}{2}\right] ^3 \cap R= & {} \left\{ (t_1,t_2,t_3) \in {{\mathbb {R}}}^3 \, : \, t_1+t_3 = \pm 1, \, -{\frac{1}{2}}\le t_2 \le {\frac{1}{2}}\right\} \\&\cup \, \, \, \left\{ (t_1,t_2,t_3) \in {{\mathbb {R}}}^3 \, : \, t_1-t_2 = \pm 1, \, -{\frac{1}{2}}\le t_3 \le {\frac{1}{2}}\right\} , \end{aligned}$$

then

$$\begin{aligned} \overline{\left[ -\frac{1}{2}, \frac{1}{2}\right] ^3 {\setminus } (\cup _{\ell =1}^6 L_{\ell }) } \cap R = \emptyset . \end{aligned}$$

Furthermore, we have that the distance between the sets R and the segment in space bounded by the points $\big ({\frac{1}{2}}, {\frac{1}{2}}, \frac{1}{4}\big )$ and $\big ({\frac{1}{2}}, {\frac{1}{2}}, -\frac{1}{2}\big )$ is the distance between the plane in space with equation $t_1+t_3=1$ and the point $\big ({\frac{1}{2}}, {\frac{1}{2}}, \frac{1}{4}\big )$, that is $\sqrt{2} / 8$. In addition, we have that the distance between the set $L_{1} + (0,1,0)$ and the segment in space bounded by the points $\big ({\frac{1}{2}}, {\frac{1}{2}}, \frac{1}{4}\big )$ and $\big ({\frac{1}{2}}, {\frac{1}{2}}, -\frac{1}{2}\big )$ is $\sqrt{2}/ 16$. This implies that

$$\begin{aligned} R \cap (L_{1} + (0,1,0)) = \emptyset . \end{aligned}$$

Similar argument can be applied to see that

$$\begin{aligned}&R \cap (L_{2} + (-1,0,0)) = \emptyset , \qquad R \cap (L_{3} + (0,0,-1)) = \emptyset , \\&R \cap (L_{4} + (0,-1,0)) = \emptyset , \qquad R \cap (L_{5} + (1,0,0)) = \emptyset , \\&R \cap (L_{6} + (0,0,1))= \emptyset . \end{aligned}$$

Thus

$$\begin{aligned} K \cap (Q^*)^{j}(Z) \subset K \cap R = \emptyset , \quad j \in {{\mathbb {N}}}. \end{aligned}$$

We conclude that $(Q^*)^{-j} K \cap Z = \emptyset$, $j \in {{\mathbb {N}}}$, that is what we wanted to see (Fig. 2).

Fig. 2

Our set K associated to $H({\mathbf {t}})$ and Q

The remaining of the proof can be done in analogous way that the proof of Proposition 2. $\square$

By Proposition 3 and Corollary 1, we have the following result.

Corollary 5

Let $a \in \{1,2\}$ and let $\varvec{\Gamma }_Q= \{ \mathbf {q}_0, \mathbf {q}_1 \}$ be a full collection of representatives of the cosets of $(Q^*)^{-1}{{\mathbb {Z}}}^{3}/ {{\mathbb {Z}}}^{3}$ with $\mathbf{q}_0=\mathbf{0}$.

Let $m, n \in {{\mathbb {N}}}$ such that $m < n$, and $k=m+1$. Let $p(t)=\widehat{a_{k,n}}(t)$ where $\widehat{a_{k,n}}$ is the univariate trigonometric polynomial defined in Theorem B.

Define

$$\begin{aligned} H(\mathbf {t}):= p (t_a), \end{aligned}$$

and functions $\phi$ and $\psi$ by

$$\begin{aligned} {\widehat{\phi }}({\mathbf {t}}):=\prod _{j=1}^{\infty } H((Q^*)^{-j}{\mathbf {t}}) \quad \text {and} \quad {\widehat{\psi }}(Q^*\mathbf{t}):= e^{2 \pi i t_a } \overline{H({\mathbf {t}} + \mathbf{q}_1)} {\widehat{\phi }}(\mathbf{t}). \end{aligned}$$

Then,

• (i) The function $\phi$ has compact support, it is symmetric respect to $\mathbf{C}= \frac{1}{2}(Q- \mathbf{I}_{3\times 3})^{-1}\mathbf {r}_1^{(a)}$ and it generates an Q-MRA.

• (ii) The function $\psi$ is a compactly supported orthonormal wavelet associated to Q. In addition, $\psi$ has vanishing moments of order m and it is antisymmetric with respect to the point $\frac{1}{2}Q^{-1}(2\mathbf{C} - 3 \mathbf{r}_1^{(a)})$.

6 Examples of Wavelets with Regularity and High Number of Vanishing Moments

Following the same strategy as in the previous section and using the unidimensional low pass filters by Daubechies in [17] (see also [18]), we construct new compactly supported wavelets with any desired degree of regularity and any fix number of vanishing moments associated to some special dilation matrices, such as $A_1$, $A_2$ or the Quincunx matrix in $\mathbb {R}^3$. This section is related to the papers by Cohen and Daubechies [15] and by Belogay and Wang [7].

We will use the following trigonometric polynomials on ${{\mathbb {R}}}$ by Daubechies in [17]. For $n=1,2,3,\ldots$, let

$$\begin{aligned} g_n(t) := 1 - c_n \int _{0}^{t}(\sin 2 \pi \xi )^{2n+1} d \xi , \end{aligned}$$

(17)

where $c_n=(\int _{0}^{1/2}(\sin 2 \pi \xi )^{2n+1} d \xi )^{-1}$. In addition, let

$$\begin{aligned} h_n(t):= \sum _{k=0}^{2n+1} \alpha _k^{(n)} e^{2 \pi i k t}, \qquad \alpha _k^{(n)} \in {{\mathbb {C}}}, \end{aligned}$$

(18)

be a trigonometric polynomial such that $|h_n(t)|^2= g_n(t)$ and $h_n(0)=1$. These polynomials $h_n$ exist by a lemma of Riesz (cf., e.g., [18, Lemma 6.1.3]), [50, Lemma 10, p. 102]). The coefficients of the polynomials $h_n$ may be obtained by spectral factorization ( [23]) and can be taken reals. Key properties of the trigonometric polynomials $h_n$ are $h_n(0)=1$,

$$\begin{aligned} |h_n(t)|^2 + |h_n(t + 1/2)|^2=1 \end{aligned}$$

(19)

and $h(t) = 0$ if and only if $t \in {\frac{1}{2}}+ {{\mathbb {Z}}}$. More properties on these trigonometric polynomials, see e.g. the textbooks [18, 34] or [56].

Indeed, we need the following result (see e.g. [56, Proposition 4.10]).

Proposition E

Let $g_n$, $n=1,2,3,\cdots$, be trigonometric polynomials defined by (17) and let us defined $G_n(t)= \prod _{j=1}^{\infty } g_n(2^{-j}t)$. Then there exist two constants $\beta >0$ and $N \in \mathbb {N}$ such that if $n >N$ and $|t|>1$ we have

$$\begin{aligned} |G_n(t)|\le C_n |t|^{-\beta n }. \end{aligned}$$

Remark 1

In Remark 4.1 of [56], it is point out that for large n one can take as $\beta$ any positive number smaller than $1- \log _2 (\frac{8}{3\sqrt{3}}) \simeq 0.3774$.

As usually in this manuscript, we start with the construction of a refinable function. Implicit in the proof of Proposition 1, it is shown the following result.

Proposition 4

Let $n \in {{\mathbb {N}}}$. Assume that there exists $S \subset {{\mathbb {R}}}^d$, a tile associated to an $A \in \mathbf{E}_d^{(2)}({{\mathbb {Z}}})$ and $\varvec{\Delta }_A= \{ \mathbf {r}_0, \mathbf {r}_1 \}$, a full collection of representatives of the cosets of ${{\mathbb {Z}}}^{d}/ A {{\mathbb {Z}}}^{d}$ with $\mathbf{r}_0=\mathbf{0}$. Let $h_n$ a trigonometric polynomial on ${{\mathbb {R}}}$ defined in (18). Let

$$\begin{aligned} H(\mathbf {t}):= h_n (\mathbf {r}_1 \cdot {\mathbf {t}}) \end{aligned}$$

(20)

and define

$$\begin{aligned} {\widehat{\phi }}({\mathbf {t}}):=\prod _{j=1}^{\infty } H((A^*)^{-j}{\mathbf {t}}). \end{aligned}$$

(21)

Then, ${\widehat{\phi }}$ is a well defined non null continuous function in $L^2({{\mathbb {R}}}^d)$ such that $\Vert {\widehat{\phi }} \Vert _{L^2({{\mathbb {R}}}^d)} \le 1$, ${\widehat{\phi }}(\mathbf{0})=1$. The function $\phi$ whose Fourier transform ${\widehat{\phi }}$ is defined by (21) has compact support and it generates an A-MRA.

From the refinable function in Proposition 4, we can construct the following wavelets.

Theorem 2

With the same hypotheses and notation as in Proposition 4, define the function $\psi \in L^2({{\mathbb {R}}}^d)$ by

$$\begin{aligned} {\widehat{\psi }}(A^*\mathbf{t}):= e^{2 \pi i \mathbf{u} \cdot \mathbf{t}} \overline{H({\mathbf {t}} + \mathbf{q}_1)} {\widehat{\phi }}(\mathbf{t}), \end{aligned}$$

(22)

where $\varvec{\Gamma }_A= \{ \mathbf {q}_0, \mathbf {q}_1 \}$ is a full collection of representatives of the cosets of $(A^*)^{-1}{{\mathbb {Z}}}^{d}/ {{\mathbb {Z}}}^{d}$ with $\mathbf{q}_0=\mathbf{0}$ and $\mathbf{u} \in {{\mathbb {Z}}}^d$ such that $\mathbf{q}_1 \cdot \mathbf{u}\in {\frac{1}{2}}+ {{\mathbb {Z}}}$. Then $\psi$ is a compactly supported orthonormal wavelet associated to A with vanishing moments of order n.

Proof

That $\psi \in L^2({{\mathbb {R}}}^d)$ defined by (22) is a compactly supported orthonormal wavelet associated to A can be proved analogously as in the proof of Theorem 1. $\square$

We are ready to construct compactly supported wavelets with high degree of regularity and vanishing moments associated to dilations integrally similar to $A_1$.

Corollary 6

Let $A \in \mathbf{E}_2^{(2)}({{\mathbb {Z}}})$ such that $A= U^{-1} A_1 U$ where U is a matrix with integer entries such that $|\det U| =1$.

Let $\mathbf{r}_1= U^{-1}({\varepsilon }_1,{\varepsilon }_2)^T$ where ${\varepsilon }_i= \pm 1$, $i=1,2$. Let $\varvec{\Gamma }_A= \{ \mathbf {q}_0, \mathbf {q}_1 \}$ be a full collection of representatives of the cosets of $(A^*)^{-1}{{\mathbb {Z}}}^{2}/ {{\mathbb {Z}}}^{2}$ with $\mathbf{q}_0=\mathbf{0}$.

Let $n \in {{\mathbb {N}}}$ and let $h_n$ be the trigonometric polynomial on ${{\mathbb {R}}}$ defined by (18). Define

$$\begin{aligned} H(\mathbf {t}):= h_n (\mathbf {r}_1 \cdot {\mathbf {t}}), \end{aligned}$$

and functions $\phi$ and $\psi$ by

$$\begin{aligned} {\widehat{\phi }}({\mathbf {t}}):=\prod _{j=1}^{\infty } H((A^*)^{-j}{\mathbf {t}}) \quad \text {and} \quad {\widehat{\psi }}(A^*\mathbf{t}):= e^{2 \pi i \mathbf{r}_1 \cdot \mathbf{t}} \overline{H({\mathbf {t}} + \mathbf{q}_1)} {\widehat{\phi }}(\mathbf{t}). \end{aligned}$$

Then the function $\psi$ is a compactly supported orthonormal wavelet associated to A with vanishing moments of order n. In addition, if $\beta \frac{n}{2} -2>\alpha >0$, then $\phi$ and $\psi$ are in continuity class $C^{\alpha }$, where $\beta$ is defined in Proposition E.

Proof

By Proposition 2, Proposition 4 and Theorem 2, we only have to prove the degree of regularity of the wavelets.

If we consider $A=A_1$, we can write

$$\begin{aligned} |{\widehat{\phi }}({\mathbf {t}})|^2= & {} \prod _{j=1}^{\infty } g_n(({\varepsilon }_1,{\varepsilon }_2)\cdot (A_1^*)^{-j} \mathbf{t}) \nonumber \\= & {} \prod _{j=1}^{\infty } g_n (2^{-j}({\varepsilon }_1 t_2 + 2 {\varepsilon }_2 t_1)) g_n (2^{-j}({\varepsilon }_1 t_1 + {\varepsilon }_2 t_2)). \end{aligned}$$

(23)

Then, if $|({\varepsilon }_1 t_2 + 2 {\varepsilon }_2 t_1)| >1$, Proposition E tells us that

$$\begin{aligned} |{\widehat{\phi }}({\mathbf {t}})|^2 \le C |({\varepsilon }_1 t_2 + 2 {\varepsilon }_2 t_1)|^{-\beta n} \end{aligned}$$

and if $|({\varepsilon }_1 t_1 + {\varepsilon }_2 t_2)|>1$,

$$\begin{aligned} |{\widehat{\phi }}({\mathbf {t}})|^2 \le C |({\varepsilon }_1 t_1 + {\varepsilon }_2 t_2)|^{-\beta n}. \end{aligned}$$

Bearing in mind that ${\widehat{\phi }}$ is continuous with ${\widehat{\phi }}(\mathbf{0})=1$, we yield to

$$\begin{aligned} |{\widehat{\phi }}({\mathbf {t}})|^2 \le {\widetilde{C}} (1 + \Vert \mathbf{t} \Vert )^{-\beta n}. \end{aligned}$$

(24)

Furthermore, since |H| is bounded above by 1, we also have

$$\begin{aligned} |{\widehat{\psi }}({\mathbf {t}})|^2 \le {\widetilde{C}} (1 + \Vert \mathbf{t} \Vert )^{-\beta n}. \end{aligned}$$

(25)

Therefore, by the well known Sobolev embedding theorems, see e.g. [20, Theorem 9.17], we conclude that both $\phi$ and $\psi$ are in continuity class $C^{\alpha }$ if $\beta \frac{n}{2} -2>\alpha >0$.

For a general $A=U^{-1}A_1 U$, the associated scaling function satisfies

$$\begin{aligned} |{\widehat{\phi }}({\mathbf {t}})|^2= & {} \prod _{j=1}^{\infty } g_n(U^{-1}({\varepsilon }_1,{\varepsilon }_2)\cdot U^*(A_1^*)^{-j}(U^{*})^{-1} \mathbf{t})\\= & {} \prod _{j=1}^{\infty } g_n(({\varepsilon }_1,{\varepsilon }_2)\cdot (A_1^*)^{-j}(U^{*})^{-1} \mathbf{t}). \end{aligned}$$

By (23) and (24), we have

$$\begin{aligned} |{\widehat{\phi }}({\mathbf {t}})|^2 \le {\widetilde{C}} (1 + \Vert (U^{*})^{-1}{} \mathbf{t} \Vert )^{-\beta n} \le \widetilde{{\widetilde{C}}} (1 + \Vert \mathbf{t} \Vert )^{-\beta n}, \end{aligned}$$

where the last inequality holds because $\det U \ne 0$. The proof is finishes as in the above case, i.e. when $A=A_1$. $\square$

More compactly supported wavelets associated to dilations integrally similar to $A_1$ or $A_2$ can be found in the following result. The proof is omitted because is similar to the proof of Corollary 6.

Corollary 7

Let $A \in \mathbf{E}_2^{(2)}({{\mathbb {Z}}})$ such that $A= U^{-1} A_i U$, $i=1,2$, where U is a matrix with integer entries such that $|\det U| =1$.

Let $\mathbf{r}_1= U^{-1}(1,0)^T$. Let $\varvec{\Gamma }_A= \{ \mathbf {q}_0, \mathbf {q}_1 \}$ be a full collection of representatives of the cosets of $(A^*)^{-1}{{\mathbb {Z}}}^{2}/ {{\mathbb {Z}}}^{2}$ with $\mathbf{q}_0=\mathbf{0}$.

Let $n \in {{\mathbb {N}}}$ and let $h_n$ be the trigonometric polynomial on ${{\mathbb {R}}}$ defined by (18). Define

$$\begin{aligned} H(\mathbf {t}):= h_n (\mathbf {r}_1 \cdot {\mathbf {t}}), \end{aligned}$$

and functions $\phi$ and $\psi$ by

$$\begin{aligned} {\widehat{\phi }}({\mathbf {t}}):=\prod _{j=1}^{\infty } H((A^*)^{-j}{\mathbf {t}}) \quad \text {and} \quad {\widehat{\psi }}(A^*\mathbf{t}):= e^{2 \pi i \mathbf{r}_1 \cdot \mathbf{t}} \overline{H({\mathbf {t}} + \mathbf{q}_1)} {\widehat{\phi }}(\mathbf{t}). \end{aligned}$$

Then the function $\psi$ is a compactly supported orthonormal wavelet associated to A with vanishing moments of order n. In addition, if $\beta \frac{n}{2} -2>\alpha >0$, then $\phi$ and $\psi$ are in continuity class $C^{\alpha }$, where $\beta$ is defined in Proposition E.

When a dilation is the Quincunx matrix in $\mathbb {R}^3$, we have the following construction of compactly supported wavelets.

Corollary 8

Let $a \in \{1,2\}$, let Q be the Quincunx matrix in $\mathbb {R}^3$ and let $\varvec{\Gamma }_Q= \{ \mathbf {q}_0, \mathbf {q}_1 \}$ be a full collection of representatives of the cosets of $(Q^*)^{-1}{{\mathbb {Z}}}^{3}/ {{\mathbb {Z}}}^{3}$ with $\mathbf{q}_0=\mathbf{0}$. Let $n \in {{\mathbb {N}}}$ and let $h_n$ be the trigonometric polynomial on ${{\mathbb {R}}}$ defined by (18).

Define

$$\begin{aligned} H(\mathbf {t}):= h_n (t_a), \end{aligned}$$

and functions $\phi$ and $\psi$ by

$$\begin{aligned} {\widehat{\phi }}({\mathbf {t}}):=\prod _{j=1}^{\infty } H((Q^*)^{-j}{\mathbf {t}}) \quad \text {and} \quad {\widehat{\psi }}(Q^*\mathbf{t}):= e^{2 \pi i t_a } \overline{H({\mathbf {t}} + \mathbf{q}_1)} {\widehat{\phi }}(\mathbf{t}). \end{aligned}$$

Then the function $\psi$ is a compactly supported orthonormal wavelet associated to Q with vanishing moments of order n. In addition, if $\beta \frac{n}{2} -2>\alpha >0$, then $\phi$ and $\psi$ are in continuity class $C^{\alpha }$, where $\beta$ is defined in Proposition E.