On the Approximation Properties of Ces\`{a}ro Means of Negative Order of double Vilenkin-Fourier Series

In this paper we establish approximation properties of Ces\`{a}ro $% (C,-\alpha ,-\beta )$ means with $\alpha ,\beta $ $\epsilon $ $(0,1)$ of Vilenkin-Fourier series.This result allows one to obtain a condition which is sufficient for the convergence of the means $\sigma _{n,m}^{-\alpha ,-\beta }(x,y,f)$ to $f(x,y)$ in the $L^{p}-$metric.

is the Haar measure on G m with µ (G m ) = 1. If the sequence m is bounded, then G m is called a bounded Vilenkin group. In this paper we will consider only bounded Vilenkin group. The elements of G m can be represented by sequences x := (x 0 , x 1 , ..., x j , ...) , x j ∈ Z m j . The group operation + in G m is given by x + y = ((x 0 + y 0 ) mod m 0 , ..., (x k + y k ) mod m k , ...) , where x := (x 0 , ..., x k , ...) and y := (y 0 , ..., y k , ...) ∈ G m . The inverse of + will be denoted by −. It is easy to give a base for the neighborhoods of G m : I 0 (x) := G m , I n (x) := {y ∈ G m |y 0 = x 0 , ..., y n−1 = x n−1 } for x ∈ G m , n ∈ N. Define I n := I n (0) for n ∈ N + . Set e n := (0, ..., 0, 1, 0, ...) ∈ G m the n + 1 th coordinate of which is 1 and the rest are zeros (n ∈ N ) . If we define the so-called generalized number system based on m in the following way: M 0 := 1, M k+1 := m k M k (k ∈ N ) , then every n ∈ N can be uniquely expressed as n = ∞ j=0 n j M j , where n j ∈ Z m j (j ∈ N + ) and only a finite number of n j 's differ from zero. We use the following notation. Let |n| :=max{k ∈ N : n k = 0} (that is , M |n| ≤ n < M |n|+1 ).
Next, we introduce of G m an orthonormal system which is called Vilenkin system. At first define the complex valued functions r k (x) : G m → C. the generalized Rademacher functions in this way Now define the Vilenkin system ψ := (ψ n : n ∈ N ) on G m as follows.
In particular, we call the system the Walsh-Paley if m = 2. The Dirichlet kernels is defined by Recall that (see [3] or [14]) The Vilenkin system is orthonormal and complete in L 1 (G m ) [1]. Next, we introduce some notation with respect to the theory of twodemonsional Vilenkin system. Letm be a sequence like m. The relation between the sequences (m n ) and M n is the same as between sequences (m n ) and (M n ) . The group G m × Gm is called a two-dimensional Vilenkin group. The normalized Haar measure is denoted by µ as in the one-dimensional case. We also suppose that m =m and G m × Gm = G 2 m . The norm of the space L p G 2 m is defined by Denote by C G 2 m the class of continuous functions on the group G 2 m , endoved with the supremum norm.
For the sake of brevity in notation, we agree to write L ∞ G 2 m instead of C G 2 m . The two-dimensional Fourier coefficients,the rectangular partial sums of the Fourier series,the Dirichlet kernels with respect to the two-dimensional Vilenkin system are defined as follow: The (c, −α, −β) means of the two-dimensional Vilenkin-Fourier series are defined as It is well Known that [18] (2) The dyadic partial moduli of continuity of a function f ∈ L p G 2 m in the L p -norm are defined by while the dyadic mixed modulus of continuity is defined as follows: it is clear that The dyadic total modulus of continuity is defined by The problems of summability of partial sums and Cesàro means for Walsh-Fourier series were studied in [2], [4]- [13], [16]. In his monography [17] Zhizhinashvili investigated the behavior of Cesàro method of negative order for trigonometric Fourier series in detail. Goginava [5] studied analogical question in case of the Walsh system. In particular, the following theorem is proved. Theorem G. [5]Let f belong to L p (G 2 ) for some p ∈ [1, ∞] and α ∈ (0, 1). Then for any 2 k ≤ n < 2 k+1 (k, n ∈ N ) the inequality The present author in [15] investigated analogous question in the case of Vilenkin system.
Theorem 2. For every α, β ∈ (0, 1) , α + β < 1, there exists a function In order to prove Theorem 1 we need the following lemmas where c is an absolute constant.
Proof of Lemma 2. Applying Abel's transformation, from (2) we get From the generalized Minkowski inequality, and by (1) and (4) we obtain It is evident that (7) It is easy to show that (8) Using Lemma 1 for I 11 we can write (9) Analogously, we can prove that (10) Combining (7)-(10) for I 1 we recive that For I 2 we can write From the generalized Minkowski inequality, and by (1) and (4) we obtain . The estimation of I 22 is analogous to the estimation of I 1 and we have (14) I 22 ≤ c (α, β) So, combining (12)- (14) for I 2 we have The estimation I 3 is analogous to the estimation of I 2 and we have Combining (5)-(6), (10), (15)- (16) we receive the proof of Lemma 2.