CONVERGENCE OF DOUBLE COSINE SERIES

In this paper we consider double cosine series whose coefficients form a null sequence of bounded variation of order (p, 0), (0, p) and (p, p) with the weight (jk)p−1 for some p > 1. We study pointwise convergence, uniform convergence and convergence in L-norm of the series under consideration. In a certain sense our results extend the results of Young [7], Kolmogorov [3] and Móricz [4, 5].


Introduction
Consider the double cosine series Now assuming the coefficients {a jk : j, k ≥ 0} in (1.1) be a double sequence of real numbers which satisfy the following conditions for some positive integer p: (1.2) |a jk |(jk) p−1 → 0 as max{j, k} → ∞, | p0 a jk |(jk) p−1 = 0, The finite order differences pq a jk are defined by 00 a jk =a jk , pq a jk = p−1,q a jk − p−1,q a j+1,k , p ≥ 1, q ≥ 0, pq a jk = p,q−1 a jk − p,q−1 a j,k+1 , p ≥ 0, q ≥ 1. Also a double induction argument gives pq a jk = p s=0 q t=0 (−1) s+t p s q t a j+s, k+t .
We can call the above mentioned conditions (1.2)-(1.5) as conditions of bounded variation of order (p, 0), (0, p) and (p, p) respectively with the weight (jk) p−1 . Obviously these conditions generalise the concept of monotone sequences. Also any sequence satisfying (1.5) with p = 2 is called a quasi-convex sequence [3,5]. λ j λ k a jk cos jx cos ky, m, n ≥ 0, are formed and then by taking both m, n tend to ∞ (independently of one another) the limit f (x, y) (provided it exists) is assigned to the series (1.1) as its sum.
Also let f r denotes the L r (T 2 )-norm, i.e, In this paper, we will investigate the validity of the following statements: Such type of problems have been studied by Young [7] and Kolmogorov [3] for onedimensional case (single trigonometric series especially cosine series ) and by Móricz [4,5] and K. Kaur, Bhatia and Ram [2] for double trigonometric series. In [5], Móricz studied both double cosine series and double sine series as far as their integrability and convergence in L 1 −norm is concerned where as in [4] he studied double trigonometric series of the form under coefficients of bounded variation. All of them discussed the case for p = 1 or p = 2 only. Our aim in this paper is to extend the above results from p = 1 to general cases for double cosine series.
In the results, C p and C pr denote constants which may not be the same at each occurrence. Also we write λ n = [λn] where n is a positive integer, λ > 1 is a real number and [·] means greatest integral part.
The first main result reads as follows.
The above theorem has been proved by Móricz [4,5] for p = 1 and p = 2 using suitable estimates for Dirichlet's kernel D j (x) and Fejér kernel K j (x). In the case of a single series for p = 2, the results regarding convergence have been proved by Kolmogorov [3].
Obviously, condition (1.5) implies any of the following conditions: We introduce the following three sums for m, n ≥ 0 and λ > 1: The second result of this paper is the following theorem.
Assume that the following conditions are satisfied: (ii) Assume that the following conditions are satisfied for some r ≥ 1: We will also prove the following theorem.

Notation and Formulas
We define for every α = 0, 1, 2, . . . the sequence S α 0 , S α 1 , S α 2 , . . . by the conditions The Cesàro means T α n of order α of a n will be defined by T α n = S α n A α n and also it is known [8] that π 0 |T α n (x)|dx, α > 0, is bounded for all n.

Lemmas
We require the following lemmas for the proof of our results.
Lemma 3.1. For m, n ≥ 0 and p > 1, the following representation holds:

Lemma 3.3.
For m, n ≥ 0 and λ > 1, we have the following representation: Performing double summation by parts, we have The use of Lemma 3.2, gives Proof. We have by summation by parts,  Similarly we can have representation for λ 01 (m, n; x, y).

Proof of Theorems
Using the results as given in [6] and also by (1.  in the following way: Now let R mn consists of all (j, k) with j > m or k > n, that is,  , as mentioned in [6], we have the following estimates in brief:   Proof of (ii). We have S mn − f r ≤ S mn − V λ mn r + V λ mn − f r . By assumption V λ mn − f r → 0, so it is sufficient to show that S mn − V λ mn r → 0 as min{m, n} → ∞. By Lemma 3.3, we have S mn − V λ mn r ≤ λ 10 (m, n; x, y) r + λ 01 (m, n; x, y) r + λ 11 (m, n; x, y) r . Now in order to estimate λ 01 (m, n; x, y) r , we first find I 1 , I 2 , I 3 , I 4 , I 5 and I 6 , so we have Thus, S mn − V λ mn r → 0 as min{m, n} → ∞.