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

We study the Walsh series expansion of multivariate functions in Lp (1 ≤ p ≤ ∞) and, in particular, in Lip(α, p). 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 ofMóricz andRhoades (1996),Nagy (2012). Our results can be applied tomany summabilitymethods, including theNörlund summability and weighted summability.

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.
Let  := {  } be a doubly infinite matrix.It is said to be doubly triangular if   = 0 for  >  or  > .
In the recent research [10], the authors established necessary conditions for a general inclusion theorem involving a pair of doubly triangular matrices.
Let   (1 ≤  ≤ ∞) denote the Lebesgue function spaces on the torus  2 ; that is,  ∈   ( 2 ).The double Walsh (Walsh-Fourier) series of such function is defined by and the Vth rectangular partial sum of  is where The th T-transformation of  V is defined by By ( 16), we have where () and  V () are the Walsh-Dirichlet kernels, in terms of  and V respectively.For any function  ∈   ( 2 ), when  is normal, Recall that the modulus of continuity of the function (, ) ∈   ( 2 ), 1-periodic in each variable, is defined by For each  > 0, the Lipschitz classes in   are defined by The (total) modulus of continuity of function (, ) ∈   ( 2 ), 1-periodic in each variable, is defined by It is easy to verify that there is a constant C > 0 such that Móricz and Siddiqi [11] studied the rate of uniform approximation by Nörlund means of Walsh (Walsh-Fourier) series of  ∈   [0, 1).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.
If {  } is nondecreasing, then one has the following estimates: The sequence {  } is called nondecreasing if it is nondecreasing in both  and ; that is, Δ 01   ≤ 0 and Δ 10   ≤ 0 for every ,  = 0, 1, . ... 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  > 0 and 1 ≤  ≤ ∞; let {  } be a double sequence of nonnegative numbers such that it is nondecreasing; Δ 11   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.

The Main Results
For a fixed ,   := {  } of nonnegative numbers tending to zero is called rest bounded variation, or briefly   ∈ RBVS, 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   ∈ HBVS, 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 0 < (  ) ≤ .
Now, we extend the concepts of RBVS and HBVS to the double sequences as follows.
It is also required that the sequence (  ) is bounded; that is, there is a positive constant  such that 0 < (  ) ≤ .
We state our main theorems as follows.

Lemma 2.
Let { V } be the same matrix as in Lemma 1; then one has (50) Meanwhile, Proof.Rewrite  2 by some different decomposition method; combining the decomposition we used in Lemma 1, it follows that (52) Furthermore, by properties ( 7) and ( 8) of   , we have We also obtain the estimate for Combining the estimates of  2 ( = 1, 2, 3, 4), we have the conclusion for  2 in Lemma 2, and the discussion for  3 is similar (as for the decomposition of  2 , we denote  3 = ∑ 4 =1  3 ).This completes the proof of Lemma 2.

Lemma 3.
Let { V } be the same matrix as in Lemma 1.
Then, one has Proof.Decompose  4 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 We denote by P  the set of Walsh polynomials of order less than ; that is, where  ≥ 1 and   denotes the real or complex numbers.On the torus  2 , we define the two-dimensional Walsh polynomials of order less than (, ) as Proof.By (6) and Hölder inequality, we have Furthermore, using the generalized Minkowski inequality, we have This completes the proof of Lemma 4.
Proof.Note that  ∈ P 2  ,2  and  ∈ P 2  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) { V } ∈ , and it is doubly triangular; then for any (ii) { V } ∈ , and it is doubly triangular; then for any Proof.(i) Since { V } is triangular, ISRN Mathematical Analysis Similarly, we have it implies that Similarly, we have This completes the proof of Lemma 6.
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.