科研之友
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摘要
For a non-negative integer n let us denote the dyadic variation of a natural number n by
V (n) := Sigma(infinity)(j=0)vertical bar n(j)-n(j+1)vertical bar+ n(0),
where n := Sigma(infinity)(i=0) n(i)2(i), n(i) is an element of {0, 1}. In this paper we prove that for a function f is an element of L log L(I-2) under the condition sup(A) V(n(A)) < infinity, the subsequence of quadratic partial sums S-nA(square) (f) of two-dimensional Walsh-Fourier series converges to the function f almost everywhere. We also prove sharpness of this result. Namely, we prove that for all monotone increasing function phi: [0, infinity) -> [0, infinity) such that phi(u) = o(u log u) as u -> infinity there exists a sequence {n(A): A >= 1} with the condition sup(A) V (n(A)) < infinity and a function f is an element of phi(L)(I-2) for which sup(A) vertical bar S-nA(square) (x(1), x(2); f)vertical bar = infinity for almost all (x(1), x(2)) is an element of I-2.
- 出版日期2018-3
