Convergence of double Fourier series and generalized Λ-variation

Abstract

The paper introduces a new concept of ƒ-variation of bivariate functions and investigates its connection with the convergence of double Fourier series.

Summability of multiple Fourier series of functions of bounded generalized variation

“…Observe that for a function f ∈ ΛBV the quadrant limits f (x ± 0, y ± 0) may not exist. As was shown in [14] for any function f ∈ Λ # BV the quadrant limits f (x ± 0, y ± 0) exist at any point (x, y) ∈ T 2 .…”

“…It is easy to show (see [7]), that n log n * BV ⊂ HBV , hence the convergence part of Theorem DW follows from Theorem S. It is essential that the condition f ∈ n log n * BV guaranties the existence of quadrant limits. The following theorem immediately follows from Theorem 1.4 and Theorem S. Theorem 2.4 (U. Goginava, A. Sahakian [14]). If Λ = {λ n } and lim sup n→∞ λ n log n n < ∞, then the class Λ # BV is a class of convergence on T 2 .…”

“…In particular, the class can not be replaced with any sequence nαn log n , where α n → ∞. Theorems 1.5, 1.6 and 2.4 imply Theorem 2.5 (U. Goginava, A. Sahakian [14]). The class B # V Φ is a class of convergence on T 2 , provided that (1.2) and (1.3) hold.…”

“…In the next theorem we characterize sequences Λ = {λ n } for which the inclusion Λ # BV ⊂ HBV holds. Definition 1.6 (U. Goginava, A. Sahakian [14]). Let Φ-be a strictly increasing continuous function on [0, +∞) with Φ (0) = 0.…”

“…(U. Goginava, A. Sahakian[14]). Let Φ and Ψ are conjugate functions in the sense of Yung (ab ≤ Φ(a) + Ψ(b)) and let…”