О частичных суммах по полиэдрам рядов Фурь е ограниченных функци й

Abstract

The following result is established. Letf be a bounded function on theN-dimensional torus, andV a polyhedron in R^N. Denote byS _nV (f) the partial sum of ordern of the Fourier series off that is generated byV. Letp∈[1, ∞). Under some natural assumptions on∂V, for every convex sequence {n _j } of integers satisfying

$$\log n_j \leqq Cj^{min (1/2N,1/pN)} , C > 0,$$

the following inequality is true:

$$\frac{1}{m}\sum\limits_{j = 1}^m {|S_{n_j V} (f,0)|^p \leqq C_1 \parallel f\parallel _\infty ^p , m = 1,2, \ldots ; C_1 = C_1 (p, N, V) > 0.} $$