Trigonometric Fourier series and Walsh-Fourier series with lacunary sequence of partial sums

Abstract

I N = {x ∈ R : 0 ≤ xj 0, (μ = μN isN-dimensional Lebesgue measure), and let f(x) = 0 on A. In the present paper, we study the behavior of the rectangular partial sums Sn(x, f ; Ψ) of multiple Fourier series with respect to the systemΨ as n → ∞, i.e.,min1≤j≤N nj → ∞, on the set A in the case where some of the components n1, . . . , nN of the vector n (the “number” of the partial sums Sn(x, f ; Ψ)) are elements of (single) lacunary sequences (a sequence {n(s)}, n(s) ∈ Z1, is said to be lacunary if n(s+1)/n(s) ≥ q > 1, s = 1, 2, . . . ). As is well known, subsequences of partial sums of Fourier series possess better properties of convergence almost everywhere (a.e.) compared to the sequence of partial sums itself. For trigonometric Fourier series, in the one-dimensional case, Kolmogorov established in [1] as early as 1922 that, for any function f ∈ L2(I), the sequence of partial sums Sn(k)(x, f ; E), where {n(k)} is a lacunary sequence, converges a.e. on I1 to f(x). This result due to Kolmogorov was extended in 1931 by Littlewood and Paley [2] to the classes Lp(I), p > 1. Later (see, for example, [3]) it was established that this result is not valid in L1(I). For a single Walsh–Fourier series, Shneider [4] found in 1950 that, for any sequence {n(k)} such that the Lebesgue constants Ln(k) are uniformly bounded (in particular, for {n(k)} = {2k}) and for any function f ∈ L1(I), the sequence of partial sums Sn(k)(x, f ;W) converges to f(x) a.e. on I1. Konyagin [5] showed in 1993 that, for the convergence a.e. of Sn(k)(x, f ;W), the boundedness condition for the Lebesgue constants is not essential. In turn, the first result for multiple trigonometric series (for N = 2) involving “lacunary sequence of partial sums”, was obtained by Sjolin in 1971 in [6], where it was proved that if f ∈ Lp(I), p > 1, and {n1 1 } is a lacunary sequence, then a.e. on I2, lim ν1, n2→∞ S n (ν1) 1 , n2 (x, f ; E) = f(x).