On the Convergence of Absolute Summability for Functions of Bounded Variation in Two Variables

and Applied Analysis 3 However, Wang and Yu 7 showed that Theorem A is not correct when 0 < α < 1. In fact, they proved the following. Theorem C. Suppose 0 < α < 1, x ∈ 0, π and f are 2π-periodic functions of bounded variation on −π,π . Then for n ≥ 2, one has Rn ( f, x ) ≤ 100 α2nα n ∑ k 1 kα−1Vπ/k 0 ( φx ) , 1.10 and there exists a 2π-periodic function f∗ of bounded variation on −π,π and a point x ∈ 0, π such that Rn ( f∗, x ) > 1 20000αnα n ∑ k 1 kα−1Vπ/k 0 ( φx ) n ≥ 8 . 1.11 Motivated by Theorems B and C, and the results of Jenei 4 and Móricz 5 on the double Fourier series, we will investigate the absolute convergence of a kind of very general summability of the double Fourier series. Our new results not only generalize Theorems B and C to the double case, but also can be applied to many other classical summability methods. We will present our main results on Section 2. Proofs will be given in Section 3. In Section 4, we will apply our results to some classical summability methods. 2. The Main Results Let f x, y be a function periodic in each variable with period 2π and integrable on the twodimensional torus T2 T × T in Lebesgue’s sense, in symbol, f ∈ L T2 . The double Fourier series of a complex-valued function f ∈ L T2 is defined by f ( x, y ) ∼ ∑ k∈Z ∑ l∈Z f̂ k, l e kx ly , 2.1 where the f̂ k, l are the Fourier coefficients of f : f̂ k, l : 1 4π2 ∫ T2 f u, v e−i ku lv dudv, k, l ∈ Z2. 2.2 Let R : a1, b1 × a2, b2 be a bounded and closed rectangle on the plane. A function f x, y defined on R is said to be of bounded variation over R in the sense of Hardy-Krause, in symbol, f x, y ∈ BVH R , if i the total variation V f, R of f x, y over R is finite, that is, V ( f, R ) : sup m ∑ j 1 n ∑ k 1 ∣f ( xj , yk ) − fxj−1,yk ) − fxj , yk−1 ) f ( xj , yk )∣ < ∞, 2.3 4 Abstract and Applied Analysis where the supremum is extended for all finite partitions a1 x0 < x1 < · · · < xm b1, a2 y0 < y1 < · · · < yn b2 2.4 of the intervals a1, b1 and a2, b2 ; ii the marginal functions f ·, a2 and f a1, · are of bounded variation over the intervals a1, b1 and a2, b2 , respectively. For any f x, y ∈ BVH T , define f ( x, y ) : 1 4 ( f ( x 0, y 0 ) f ( x − 0, y 0 fx 0, y − 0 fx − 0, y − 0, φxy u, v : f ( x u, y v ) f ( x − u, y v fx u, y − v fx − u, y − v − 4fx, y. 2.5 For convenience, write V st 00 ( φxy ) : V ( φxy, 0, s × 0, t ) . 2.6 For any double sequence {amn}, define Δ11amn : amn − am−1,n − am,n−1 am−1,n−1, Δ10amn : amn − am−1,n, Δ01amn : amn − am,n−1. 2.7 For any fourfold sequence {amnjk}, write Δ11amnjk : amnjk − am,n,j 1,k − am,n,j,k 1 am,n,j 1,k 1, Δ01amnjk : amnjk − am,n,j,k 1, Δ10amnjk : am,n,j,k − am,n,j 1,k. 2.8 A doubly infinite matrix T : tmnjk is said to be doubly triangular if tmnjk 0 for j > m or k > n. Themnth term of the T -transform of the double Fourier series 2.1 is defined by Tmn ( x, y ) : m ∑ μ 0 n ∑ ν 0 tmnμνSμν ( x, y ) , 2.9 where Sμν x, y is the μνth partial sum of 2.1 , that is, Sμν ( x, y ) : μ ∑ |k| 0 ν ∑ |l| 0 f̂ k, l e kx ly : μ ∑ |k| 0 ν ∑ |l| 0 Akl ( x, y ) . 2.10 Abstract and Applied Analysis 5 Writeand Applied Analysis 5 Write