Uniform convergence of Cesàro means of negative order of double trigonometric Fourier series

Abstract

In this paper we prove that if f ∊ C([−π, π]2) and the function f is bounded partial p-variation for some p ∊ [1,+ ∞), then the double trigonometric Fourier series of a function f is uniformly (C;−α,−β) summable (α+β> 1/p, α, β> 0) in the sense of Pringsheim. If α + β ≥ 1/p, then there exists a continuous function f0 of bounded partial p-variation on [−π, π]2 such that the Cesàro (C;−α,−β) means σ−α,−βn,m (f0;0,0) of the double trigonometric Fourier series of f0 diverge over cubes.