## Abstract

For a non-negative integer n let us denote the dyadic variation of a natural number n by V(n):=∑j=0∞|nj−nj+1|+n0, where n := ∑_{i=0}^{∞}n_{i}2^{i}, n_{i} ∈ {0, 1}. In this paper we prove that for a function f ∈ L log L(I^{2}) under the condition sup_{A}V (nA) < ∞, the subsequence of quadratic partial sums Sn□A(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 φ: [0,∞) → [0,∞) such that φ(u) = o(u log u) as u → ∞ there exists a sequence {n_{A} : A ≥ 1} with the condition sup_{A}V(nA) < ∞ and a function f ∈ φ(L)(I^{2}) for which supA|Sn◻A(x1,x2;f)|=∞ for almost all (x^{1}, x^{2}) ∈ I^{2}.

Original language | English |
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Pages (from-to) | 73-88 |

Number of pages | 16 |

Journal | Analysis Mathematica |

Volume | 44 |

Issue number | 1 |

DOIs | |

Publication status | Published - Mar 1 2018 |

Externally published | Yes |

## ASJC Scopus subject areas

- General Mathematics