Matrix Summability of Fourier and Walsh-Fourier Series

Abstract

In this chapter we apply regular and almost regular matrices to find the sum of derived Fourier series, conjugate Fourier series, and Walsh-Fourier series (see [4] and [69]). Recently, Móricz [67] has studied statistical convergence of sequences and series of complex numbers with applications in Fourier analysis and summability.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

8.1 Introduction

In this chapter we apply regular and almost regular matrices to find the sum of derived Fourier series, conjugate Fourier series, and Walsh-Fourier series (see [4] and [69]). Recently, Móricz [67] has studied statistical convergence of sequences and series of complex numbers with applications in Fourier analysis and summability.

8.2 Summability of Fourier Series

Let f be L-integrable and periodic with period 2π, and let the Fourier series of f be

$$\displaystyle\begin{array}{rcl} \frac{1} {a_{0}} +\sum _{ k=1}^{\infty }\left (a_{ k}\cos kx + b_{k}\sin kx\right ).\end{array}$$

(8.2.1)

Then, the series conjugate to it is

$$\displaystyle\begin{array}{rcl} \sum _{k=1}^{\infty }\left (b_{ k}\cos kx - a_{k}\sin kx\right ),& &{}\end{array}$$

(8.2.2)

and the derived series is

$$\displaystyle\begin{array}{rcl} \sum _{k=1}^{\infty }k\left (b_{ k}\cos kx - a_{k}\sin kx\right ).& &{}\end{array}$$

(8.2.3)

Let S _n (x), $\tilde{S}_{n}(x)$, and $S_{n}^{{\prime}}(x)$ denote the partial sums of series (8.2.1), (8.2.2), and (8.2.3), respectively. We write

$$\displaystyle\begin{array}{rcl} \psi _{x}(t) =\psi (f,t) = \left \{\begin{array}{ccl} f(x + t) - f(x - t)&,&0 < t \leq \pi;\\ g(x), & &t = 0 \end{array} \right.& & {}\\ \end{array}$$

and

$$\displaystyle\begin{array}{rcl} \beta _{x}(t) = \frac{\psi _{x}(t)} {4\sin \frac{1} {2}t},& & {}\\ \end{array}$$

where $g(x) = f(x + 0) - f(x - 0)$. These formulae are correct a.e..

Theorem 8.2.1.

Let f(x) be a function integrable in the sense of Lebesgue in [0,2π] and periodic with period 2π. Let A = (a _nk ) be a regular matrix of real numbers. Then for every x ∈ [−π,π] for which β _x (t) ∈ BV [0,π],

$$\displaystyle\begin{array}{rcl} \lim _{n\rightarrow \infty }\sum _{k=1}^{\infty }a_{ nk}S_{k}^{{\prime}}(x) =\beta _{ x}(0+)& &{}\end{array}$$

(8.2.4)

if and only if

$$\displaystyle\begin{array}{rcl} \lim _{n\rightarrow \infty }\sum _{k=1}^{\infty }a_{ nk}\sin \left (k + \frac{1} {2}\right )t = 0& &{}\end{array}$$

(8.2.5)

for every t ∈ [0,π], where BV [0,π] denotes the set of all functions of bounded variations on [0,π].

We shall need the following well-known Dirichlet-Jordan Criterion for Fourier series [101].

Lemma 8.2.2 (Dirichlet-Jordan Criterion for Fourier Series).

The trigonometric Fourier series of a 2π-periodic function f having bounded variation converges to $[f(x + 0) - f(x - 0)]/2$ for every x and this convergence is uniform on every closed interval on which f is continuous.

We shall also need the following result on the weak convergence of sequences in the Banach space of all continuous functions defined on a finite closed interval [11].

Lemma 8.2.3.

Let C[0,π] be the space of all continuous functions on [0,π] equipped with the sup-norm $\|.\|.$ Let g _n ∈ C[0,π] and $\int _{0}^{\pi }g_{n}dh_{x} \rightarrow 0$ , as n →∞, for all h _x ∈ BV [0,π] if and only if $\|g_{n}\| < \infty$ for all n and g _n → 0, as n →∞.

Proof.

We have

$$\displaystyle\begin{array}{rcl} S_{k}^{{\prime}}(x)& =& \frac{1} {\pi } \int _{0}^{\pi }\psi _{ x}(t)\left (\sum _{m=1}^{k}m\sin mt\right )dt {}\\ & =& -\frac{1} {\pi } \int _{0}^{\pi }\psi _{ x}(t) \frac{d} {dt}\left [\frac{\sin \left (k + \frac{1} {2}\right )t} {2\sin \frac{t} {2}} \right ]dt {}\\ & =& I_{k} + \frac{2} {\pi } \int _{0}^{\pi }\sin \left (k + \frac{1} {2}\right )td\beta _{x}(t), {}\\ \end{array}$$

where

$$\displaystyle\begin{array}{rcl} I_{k} = \frac{1} {\pi } \int _{0}^{\pi }\beta _{ x}(t)\cos \frac{t} {2}\left [\frac{\sin \left (k + \frac{1} {2}\right )t} {\sin \frac{t} {2}} \right ]dt.& & {}\\ \end{array}$$

Then,

$$\displaystyle\begin{array}{rcl} \sum _{k=1}^{\infty }a_{ nk}S_{k}^{{\prime}}(x) =\sum _{ k=1}^{\infty }a_{ nk}I_{k} + \frac{2} {\pi } \int _{0}^{\pi }L_{ n}(t)\text{ }d\beta _{x}(t),& & {}\\ \end{array}$$

where

$$\displaystyle\begin{array}{rcl} L_{n}(t) =\sum _{ k=1}^{\infty }a_{ nk}\sin \left (k + \frac{1} {2}\right )t.& & {}\\ \end{array}$$

Since β _x (t) is of bounded variation on [0, π] and β _x (t) → β _x (0+) as t → 0, $\beta _{x}(t)\cos \frac{t} {2}$ has also the same properties. Hence, by Lemma 8.2.2, $I_{k} \rightarrow \beta _{x}(0+)$ as k → ∞.

Since the matrix A = (a _nk ) is regular, we have

$$\displaystyle\begin{array}{rcl} \lim _{n\rightarrow \infty }\sum _{k=1}^{\infty }a_{ nk}I_{k} =\beta _{x}(0+).& &{}\end{array}$$

(8.2.6)

Now, it is enough to show that (8.2.5) holds if and only if

$$\displaystyle\begin{array}{rcl} \lim _{n\rightarrow \infty }\int _{0}^{\pi }L_{ n}(t)\text{ }d\beta _{x}(t) = 0.& &{}\end{array}$$

(8.2.7)

Hence, by Lemma 8.2.3, it follows that (8.2.7) holds if and only if

$$\displaystyle\begin{array}{rcl} \|L_{n}(t)\text{ }\| \leq M\text{ for all }n\text{ and for all }t \in [0,\pi ],& &{}\end{array}$$

(8.2.8)

and (8.2.5) holds, where M is a constant. Since (8.2.8) is satisfied by the regularity of A, it follows that (8.2.7) holds if and only if (8.2.5) holds. Hence the result follows immediately.

This completes the proof. □ 

Similarly we can prove the following result for almost regularity.

Theorem 8.2.4.

Let f be a function integrable in the sense of Lebesgue in [0,2π] and periodic with period 2π. Let A = (a _nk ) be an almost regular matrix of real numbers. Then for every x ∈ [−π,π] for which β _x (t) ∈ BV [0,π],

$$\displaystyle\begin{array}{rcl} \lim _{p\rightarrow \infty } \frac{1} {p + 1}\sum _{j=n}^{n+p}\sum _{ k=1}^{\infty }a_{ jk}S_{k}^{{\prime}}(x) =\beta _{ x}(0+)\text{ uniformly in }n& & {}\\ \end{array}$$

if and only if

$$\displaystyle\begin{array}{rcl} \lim _{p\rightarrow \infty } \frac{1} {p + 1}\sum _{j=n}^{n+p}\sum _{ k=1}^{\infty }a_{ jk}\sin \left (k + \frac{1} {2}\right )t = 0\text{ uniformly in }n& & {}\\ \end{array}$$

for every t ∈ [0,π].

Theorem 8.2.5.

Let f(x) be a function integrable in the sense of Lebesgue in [0,2π] and periodic with period 2π. Let A = (a _nk ) be a regular matrix of real numbers. Then A-transform of the sequence $\{k\tilde{S}_{k}(x)\}$ converges to g(x)∕π, i.e.,

$$\displaystyle\begin{array}{rcl} \lim _{n\rightarrow \infty }\sum _{k=1}^{\infty }ka_{ nk}\tilde{S}_{k}(x) = \frac{1} {\pi } g(x)& &{}\end{array}$$

(8.2.9)

if and only if

$$\displaystyle\begin{array}{rcl} \lim _{n\rightarrow \infty }\sum _{k=0}^{\infty }a_{ nk}\cos kt = 0& &{}\end{array}$$

(8.2.10)

for every t ∈ (0,π], where each $a_{k},b_{k} \in BV [0,2\pi ].$

Proof.

We have

$$\displaystyle\begin{array}{rcl} \tilde{S}_{n}(x)& =& \frac{1} {\pi } \int _{0}^{\pi }\psi _{ x}(t)\sin nt\text{ }dt, {}\\ & =& \frac{g(x)} {n\pi } + \frac{1} {n\pi }\int _{0}^{\pi }\cos nt\text{ }d\psi _{ x}(t). {}\\ \end{array}$$

Therefore

$$\displaystyle\begin{array}{rcl} \sum _{k=1}^{\infty }ka_{ nk}\tilde{S}_{k}(x) = \frac{g(x)} {\pi } \sum _{k=1}^{\infty }a_{ nk} + \frac{1} {\pi } \int _{0}^{\pi }K_{ n}(t)\text{ }d\psi _{x}(t),& &{}\end{array}$$

(8.2.11)

where

$$\displaystyle\begin{array}{rcl} K_{n}(t) =\sum _{ k=1}^{\infty }a_{ nk}\cos kt.& & {}\\ \end{array}$$

Now, taking limit as n → ∞ on both sides of (8.2.10) and using Lemma 8.2.3 and regularity conditions of A as in the proof of Theorem 8.2.1, we get the required result. □ 

Remark 8.2.6.

Analogously, we can state and prove Theorem 8.2.4 for almost regular matrix A.

8.3 Summability of Walsh-Fourier Series

Let us define a sequence of functions $h_{0}(x),h_{1}(x),\ldots,h_{n}(x)$ which satisfy the following conditions:

$$\displaystyle\begin{array}{rcl} h_{0}(x) = \left \{\begin{array}{ccl} 1, &&0 \leq x \leq \frac{1} {2}, \\ - 1,&&\frac{1} {2} \leq x < 1, \end{array} \right.& & {}\\ \end{array}$$

$h_{0}(x + 1) = h_{0}(x)$ and $h_{n}(x) = h_{0}({2}^{n}x),$ n = 1, 2, …. The functions h _n (x) are called the Rademacher’s functions.

The Walsh functions are defined by

$$\displaystyle\begin{array}{rcl} \phi _{n}(x) = \left \{\begin{array}{ccl} 1, &&n = 0, \\ h_{n_{1}}(x)h_{n_{2}}(x)\cdots h_{n_{r}}(x),&&n > 1,\ 0 \leq x \leq 1 \end{array} \right.& & {}\\ \end{array}$$

for $n = {2}^{n_{1}} + {2}^{n_{2}} + \cdots + {2}^{n_{r}},$ where the integers n _i are uniquely determined by $n_{i+1} < n_{i}$.

Let us recall some basic properties of Walsh functions (see [34]). For each fixed x ∈ [0, 1) and for all t ∈ [0, 1)

• (i) $\phi _{n}(x\dot{ +} t) =\phi _{n}(x)\phi _{n}(t),$

• (ii) $\int _{0}^{1}f(x\dot{ +} t)dt =\int _{ 0}^{1}f(t)dt,$ and

• (iii) $\int _{0}^{1}f(t)\phi _{n}(x\dot{ +} t)dt =\int _{ 0}^{1}f(x\dot{ +} t)\phi _{n}(t)dt,$

where $\dot{+}$ denotes the operation in the dyadic group, the set of all sequences s = (s _n ), s _n  = 0, 1 for n = 1, 2, … is addition modulo 2 in each coordinate.

Let for x ∈ [0, 1),

$$\displaystyle\begin{array}{rcl} J_{k}(x) =\int _{ 0}^{x}\phi _{ k}(t)dt,\text{ }k = 0,1,2,\ldots & & {}\\ \end{array}$$

It is easy to see that J _k (x) = 0 for x = 0, 1. 

Let f be L-integrable and periodic with period 1, and let the Walsh-Fourier series of f be

$$\displaystyle\begin{array}{rcl} \sum _{n=1}^{\infty }c_{ n}\phi _{n}(x),& & {}\\ \end{array}$$

where

$$\displaystyle\begin{array}{rcl} c_{n} =\int _{ 0}^{1}f(x)\phi _{ n}(x)dx& & {}\\ \end{array}$$

are called the Walsh-Fourier coefficients of f.

The following result is due to Siddiqi [91].

Theorem 8.3.1.

Let A = (a _nk ) be a regular matrix of real numbers. Let $z_{k}(x) = c_{k}\phi _{k}(x)$ for an L-integrable function f ∈ BV [0,1). Then for every x ∈ [0,1)

$$\displaystyle\begin{array}{rcl} \lim _{n\rightarrow \infty }\sum _{k=1}^{\infty }a_{ nk}z_{k}(x) = 0& & {}\\ \end{array}$$

if and only if

$$\displaystyle\begin{array}{rcl} \lim _{n\rightarrow \infty }\sum _{k=1}^{\infty }a_{ nk}J_{k}(x) = 0,& & {}\\ \end{array}$$

where x is a point at which f(x) is of bounded variation.

This can be proved similarly as our next result which is due to Mursaleen [69] in which we use the notion of F _A -summability. Recently, Alghamdi and Mursaleen [4] have applied Hankel matrices for this purpose.

Theorem 8.3.2.

Let A = (a _nk ) be a regular matrix of real numbers. Let $z_{k}(x) = c_{k}\phi _{k}(x)$ for an L-integrable function f ∈ BV [0,1). Then for every x ∈ [0,1), the sequence {z _k (x)} _k is F _A -summable to 0 if and only if the sequence $\{J_{k}(x)\}_{k}$ is F _A -summable to 0, that is,

$$\displaystyle\begin{array}{rcl} \lim _{n\rightarrow \infty }\sum _{k=1}^{\infty }a_{ nk}z_{k+p}(x) = 0,\text{ uniformly in }p& & {}\\ \end{array}$$

if and only if

$$\displaystyle\begin{array}{rcl} \lim _{n\rightarrow \infty }\sum _{k=1}^{\infty }a_{ nk}J_{k}(x) = 0\text{ uniformly in }p,& & {}\\ \end{array}$$

where x is a point at which f(x) is of bounded variation.

Proof.

We have

$$\displaystyle\begin{array}{rcl} z_{k}(x)& =& c_{k}\phi _{k}(x) =\int _{ 0}^{1}f(t)\phi _{ k}(t)\phi _{k}(x)dt, {}\\ & =& \int _{0}^{1}f(t)\phi _{ k}(x\dot{ +} t)dt =\int _{ 0}^{1}f(x\dot{ +} t)\phi _{ k}(t)dt, {}\\ \end{array}$$

where $x\dot{ +} t$ belongs to the set $\Omega$ of dyadic rationals in [0, 1); in particular each element of $\Omega$ has the form p∕2ⁿ for some nonnegative integers p and n, 0 ≤ p < 2ⁿ. Now, on integration by parts, we obtain

$$\displaystyle\begin{array}{rcl} z_{k}(x)& =& [f(x\dot{ +} t)J_{k}(t)]_{0}^{1} -\int _{ 0}^{1}J_{ k}(t)df(x\dot{ +} t), {}\\ & =& -\int _{0}^{1}J_{ k}(t)df(x\dot{ +} t),\text{ since }J_{k}(x) = 0\ \text{ for }\ x \in \{ 0,1\}. {}\\ \end{array}$$

Hence, for a regular matrix A = (a _nk ) and p ≥ 0, we have

$$\displaystyle\begin{array}{rcl} \sum _{k=1}^{\infty }a_{ nk}z_{k+p}(x) = -\int _{0}^{1}D_{ np}(t)\text{ }dh_{x}(t),& &{}\end{array}$$

(8.3.1)

where

$$\displaystyle\begin{array}{rcl} D_{np}(t) =\sum _{ k=1}^{\infty }a_{ nk}J_{k+p}(t),& &{}\end{array}$$

(8.3.2)

and $h_{x}(t) = f(x\dot{ +} t)$. Write, for any $t \in \mathbb{R},$ $g_{np} = (D_{np}(t))$.

Since A is regular (and hence almost regular), it follows that $\|g_{np}\| < \infty$ for all n and p, and g _np  → 0, as n → ∞ pointwise, uniformly in p. Hence by Lemma 8.2.3,

$$\displaystyle\begin{array}{rcl} \int _{0}^{1}D_{ np}(t)dh_{x}(t) \rightarrow 0& & {}\\ \end{array}$$

as n → ∞ uniformly in p. Now, letting n → ∞ in (8.3.1) and (8.3.2) and using Lemma 8.2.3, we get the desired result.

This completes the proof. □ 

Remark 8.3.3.

If we take the matrix A as the Cesàro matrix (C, 1), then we get the following result for almost summability.

Theorem 8.3.4.

Let A = (a _nk ) be almost regular matrix of real numbers. Let $z_{k}(x) = c_{k}\phi _{k}(x)$ for an L -integrable function f ∈ BV [0,1). Then for every x ∈ [0,1)

$$\displaystyle\begin{array}{rcl} F -\lim _{n\rightarrow \infty }\sum _{k=1}^{\infty }a_{ nk}z_{k}(x) = 0& & {}\\ \end{array}$$

if and only if

$$\displaystyle\begin{array}{rcl} F -\lim _{n\rightarrow \infty }\sum _{k=1}^{\infty }a_{ nk}J_{k}(x) = 0\text{,}& & {}\\ \end{array}$$

where x is a point at which f is of bounded variation.