On the convergence of the Abel–Poisson means of multiple Fourier series

Abstract

Let Aε(x, f) be the Abel–Poisson means of an integrable function f(x) on n–dimensional torus $T^n$ , −π < xi ≤ π, i = 1, . . . , n (n ≥ 2) in the Euclidean n–space. The famous Bochner’s theorem asserts that for any function f ∈ L 1 (T n ) the Abel–Poisson means Aε(x, f) are pointwise converge to f(x) a.e., that is, lim ε→0+ Aε(x, f) = f(x), a.e. x ∈ $T^n$ . In this paper we investigate the rate of convergence of Abel–Poisson means at the so-called µ–smoothness point of f .