Trigonometric Fourier Series and Walsh-Fourier Series with Lacunary Sequence of Partial Sums

摘要

Let the function f(x) belonging to L_1 (Ⅱ~N), N ≥ 1, where Ⅱ~N = {x ∈ R~N : 0 ≤ x_j ＜ 1, j = 1,…, N}, be expanded in a multiple Fourier series with respect to the trigonometric system ψ = ε and in a multiple Fourier series with respect to the Walsh-Paley system ψ = W. Also let S_n(x,f;ε), S_n(x,f;W), n = (n_1,…,n_N) ∈ Z_1~N ={n ∈ Z~N :n_j≥1, j=1,…, N}, be the rectangular partial sums of these series.