Approximation by 𝑇-Transformation of Double Walsh-Fourier Series to Multivariable Functions

Abstract

We study the Walsh series expansion of multivariate functions in and, in particular, in . The rate of uniform approximation by T-transformation of rectangular partial sums of double Walsh to these functions is investigated. By extending the concepts of rest (head) bounded variation series, which was introduced by Leindler (2004), we generalize the related results of Móricz and Rhoades (1996), Nagy (2012). Our results can be applied to many summability methods, including the Nörlund summability and weighted summability.

1. Introduction

Let be the function defined on by

The Rademacher system is defined by

Let be the Walsh functions, where , with the dyadic expansion and , respectively; here for . We also write . The idea of using products of Rademacher’s functions to construct the Walsh system originated from Paley [1].

As an important orthonormal bases, Walsh functions have most of the properties of Fourier series but are more suited to nonlinear studies. If a zero-memory nonlinear transformation is applied to a Walsh series, the output series can be derived by simple algebraic processes. Corrington [2] proved that nonlinear differential and integral equations can be solved by Walsh series. Meanwhile, the Walsh functions are of great practical interest. They have many applications in signal processing [3], dynamic systems, identification, control [4, 5], and so on.

In the above-mentioned issues, Walsh series expansion of certain function and its convergence to that function play very important roles. In this paper, we are interested in the Walsh expansion of multivariate functions and discuss the convergence of its T-transformation to these functions (we mention here that, in order to avoid notational difficulties, we restrict ourselves to the case of bivariate functions). Furthermore, the results are applied to some summability methods.

The Walsh-Dirichlet kernels and Walsh-Fejér kernels are defined by

It is known that [6] where denotes the dyadic interval in defined by , and , .

In addition, we point out that the standard representations for the Walsh-Dirichlet kernel [7, 8] are

For the Walsh-Fejér kernel , let ; Yano [9] proved that

Let be a doubly infinite matrix. It is said to be doubly triangular if for or . In the recent research [10], the authors established necessary conditions for a general inclusion theorem involving a pair of doubly triangular matrices.

Given a double sequence of complex numbers, the th term of the -transformation of is defined by

If , then we say that is normal.

Let be a double sequence of nonnegative numbers; . Taking where . Then the corresponding is known as the Nörlund means. The Cesàro summability of orders , denoted by , is a special case of the Nörlund summability with for , and . In this case,

If we take the corresponding is the well-known Riesz means of .

Let () denote the Lebesgue function spaces on the torus ; that is, . The double Walsh (Walsh-Fourier) series of such function is defined by and the th rectangular partial sum of is where

The th T-transformation of is defined by

By (16), we have where and are the Walsh-Dirichlet kernels, in terms of and respectively.

For any function , when is normal,

Recall that the modulus of continuity of the function -periodic in each variable, is defined by

For each , the Lipschitz classes in are defined by

The (total) modulus of continuity of function , -periodic in each variable, is defined by

It is easy to verify that there is a constant such that

Móricz and Siddiqi [11] studied the rate of uniform approximation by Nörlund means of Walsh (Walsh-Fourier) series of . Later, Móricz and Rhoades [12] studied the corresponding approximation problem by weighted means of Walsh-Fourier series. Their main results in [12] can be read as follows.

Theorem A. Let .
(i) If   is nondecreasing and satisfies the condition then
(ii) If   is nonincreasing, then

Theorem B. Let for and . If   is nondecreasing, then one has the following estimates:

For any fourfold sequence , write

The sequence is called nondecreasing if it is nondecreasing in both and ; that is, and for every . The nonincreasing case is defined analogously.

Recently, Nagy [13] did some research on the approximation by Nörlund means of double Walsh-Fourier series for Lipschitz functions and generalized Theorems A and B to the functions of two variables. We present one of the main results in [13] here.

Theorem C. Let for some and ; let be a double sequence of nonnegative numbers such that it is nondecreasing; is of fixed sign and satisfies the regularity condition: then, where and   is defined as in (16).

We know that in the theory of Fourier series it is of main interest how to approximate the function from the partial sums of its Fourier series. The purpose of the present paper is to get the rate of uniform approximation by -transformation with doubly triangular. We give the outline of the paper. In Section 2, we state the main results. Some auxiliary lemmas are given in Section 3, and the proofs of the main theorems are presented in Section 4. Our new results can be applied to many classical summability methods such as Nörlund summability and Riesz summability. As an important application, we will apply them to the Nörlund summability and weighted means in Section 5. We will see that not only Theorems A, B, and C are corollaries of our results but also some other new types of estimates are presented in this paper.

2. The Main Results

For a fixed , of nonnegative numbers tending to zero is called rest bounded variation, or briefly , if there is a constant , only depending on , such that holds for all natural numbers .

For a fixed of nonnegative numbers tending to zero is called head bounded variation, or briefly , if there is a constant only depending on such that for all natural numbers , or only for all if the sequence has only finite nonzero terms, and the last nonzero term is .

Remark A. The definitions of RBVS and HBVS are introduced by Leindler [14] to generalize the monotonicity conditions on sequences. In fact, RBVS and HBVS generalized monotone nonincreasing sequences and monotone nondecreasing sequences, respectively.

Remark B. Since it involves a sequence , there should be a constant such that .

Now, we extend the concepts of RBVS and HBVS to the double sequences as follows.

Definition 1. A double sequence is called DRBVS, if there is a constant such that for ,

Definition 2. A double sequence is called DHBVS, if there is a constant such that for ,

It is also required that the sequence is bounded; that is, there is a positive constant such that .

We state our main theorems as follows.

Theorem 3. Let , be a a doubly normal triangular matrix of nonnegative numbers, DRBVS, and let . Then, one has
Also, if , is doubly normal triangular of nonnegative numbers; DHBVS; then one has the conclusion as follows:

Theorem 4. Suppose that for . Let be nonnegative, doubly normal triangular, and DHBVS, satisfying  . Then, one has While for and for , let be nonnegative, normal doubly triangular, and DRBVS; one has

3. Lemmas and Proofs

Lemma 1. Suppose that is doubly normal triangular and is defined as (4); write ; then one has

Proof. We can rewrite as and keep dividing the above into 4 parts:
By using (8), we have
Applying double Abel’s transformation,
Next, (5) leads to
Using (8), (5), and Abel’s transformation, similar to the estimate of (more easily actually),
Analogously,
Combining the above estimates of , we obtain the conclusion of Lemma 1.

Lemma 2. Let be the same matrix as in Lemma 1; then one has Meanwhile,

Proof. Rewrite by some different decomposition method; combining the decomposition we used in Lemma 1, it follows that
Furthermore, by properties (7) and (8) of , we have
By changing to and applying Abel’s transformation and (5),
Similarly,
We also obtain the estimate for
Combining the estimates of , we have the conclusion for in Lemma 2, and the discussion for is similar (as for the decomposition of , we denote ). This completes the proof of Lemma 2.

Lemma 3. Let be the same matrix as in Lemma 1. Then, one has

Proof. Decompose into 4 parts:
Using the techniques as in Lemmas 1 and 2, we can easily get the result of Lemma 3.
By Lemmas 1, 2, and 3 and (20), can be written as
Therefore, by (21), we have
We denote by the set of Walsh polynomials of order less than ; that is, where and denotes the real or complex numbers. On the torus , we define the two-dimensional Walsh polynomials of order less than as

Lemma 4. Consider

Proof. By (6) and Hölder inequality, we have
Furthermore, using the generalized Minkowski inequality, we have
This completes the proof of Lemma 4.

Lemma 5. Let , ; then one has(i) (ii)

Proof. Note that and are constants on the sets and , respectively. By Lemma 4, it is not difficult to prove Lemma 5. We can also find the conclusions in [13].

Lemma 6. Suppose that (i), and it is doubly triangular; then for any or , one has or .(ii), and it is doubly triangular; then for any or , one has or .

Proof. (i) Since is triangular, Similarly, we have
(ii) Since it implies that
Similarly, we have
This completes the proof of Lemma 6.

4. Proofs of Theorems 3 and 4

We denote the norm of by Then by (60) and Minkowski inequality, we have

Proof of Theorem 3. (I) When , we deduce separately, by (9) and Lemma 5(i), Since , by the definition of , we have Substituting the above into yields Now by (9) and Lemma 5(ii), Applying Lemma 6(i) and still the definition of , we have Analogously, By Lemmas 4 and 6(i), it is easy to verify that Similar discussion deduces that (by using Lemmas 4 and 5 (i) and (ii) separately)
Note the definition of and ; clearly, and ; thus we have
Applying inequality (83) and Lemma 6 to (82), also using the monotonicity of continuous modulus, it is not difficult to deduce that
goes analogously as ; thus we have
Next, we estimate . Applying Lemmas 4 and 6, we easily get and the last term
Applying Lemma 6(i) and (83) to (86) and (87), we have
Combining all the estimates, (77)–(81), (84), (85), and (88) of , yields that
(II) When , the proof goes similarly. We mainly point out the differences in this case and prove some terms of as examples.
Since , by the definition of , we have
Thus by Lemma 5(i), (9), (71), and (72), we have
Lemma 5(ii), (9), (71), (72), and (83) yield that
Now the definition of , Lemmas 5 and 6, and (9) deduce that
While in this case we keep in the sum, so we have
Analogously,
Combining all the estimates of (92)–(96), we obtain the conclusion of the case of in Theorem 3.

Remark C. Actually, in the case of , (96) can be represented in a more complicated form if we calculate the term with the same manner. For instance, while we give a simpler form in Theorem 3 of the present paper.

Proof of Theorem 4. Since , and , in the case of , it is easy to calculate that
Note that and ; the above estimate is equivalent to
While for , the conclusion is obvious from Theorem 3 and the supposed .

Comparing Theorems 3 and 4 with Theorems A, B, and C, we find that the former are the generalizations of the latter, from the sense of monotonicity on one hand. On the other hand, -transformation is the generalization of some means of series, such as Nörlund means, Cesàro means, and Riesz means. In the next section, we give applications of our results to some summability methods.

5. Applications to Nörlund Means and Weighted Means

Corollary 5. Let be the matrix defined by (11). Suppose that , for . Then, one has
For , one has

Proof. Note that in the case when , we have . By the first part of Theorem 3, it is easy to deduce that
When (corresponding to the case ), we get the conclusion from the second part of Theorem 3 immediately.

Remark D. When is nondecreasing and is of fixed sign, it is obvious that . Therefore, Theorem 3 of [13] can be deduced directly from the first inequality of Corollary 5. Furthermore, we exclude some preconditions in this case. For the nonincreasing case of Theorem in [13], as we figure out in Remark C, the second inequality of Corollary 5 can be seen as the generalization of this case as long as it takes the more complicated form of , .

The following is a Corollary of Theorem 4, for the case of Nörlund means. Since the nondecreasing of and fixed sign of imply that , it is the generalization of Theorem C.

Corollary 6. Let be the matrix defined by (11). Suppose that satisfies
Then for , one has

As far as the weighted means is concerned, we have the following corollary which generalizes Theorems A and B.

Corollary 7. Let be the matrix defined by (14) and . Suppose that . Then, one has (101). For , one has (100).
Suppose that , satisfies
Then, one has

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.