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F. Schipp in 1969 proved the almost everywhere p-strong summability of Walsh-Fourier series and showed that if λ( n) →∞, then there exists a function f ∈ L[0, 1) for which the Walsh partial sums S( x, f) satisfy the divergence condition almost everywhere on [0, 1). In the present paper, we show that this condition holds everywhere. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the structural and geometric characteristics of some subsets of $$ \mathbb{T} $$ N = [ −π, π] N (of positive measure), on which, for the classes L p ( $$ \mathbb{T} $$ N ), p > 1, where N ≥ 3, weak generalized localization for multiple trigonometric Fourier series is valid almost everywhere, provided that the rectangular partial sums S n ( x; f) ( x ∈ $$ \mathbb{T} $$ N , f ∈ L p ) of these series have a “number” n = ( n 1,..., n N ) ∈; ℤ such that some components n j are elements of lacunary sequences. For N = 3, similar studies are carried out for generalized localization almost everywhere. [ABSTRACT FROM AUTHOR]

We construct an example of a function from the class $H\_1^\{\backslash omega^*\}$, where $\backslash omega^*(t)=\backslash sqrt\{\backslash log\backslash log(t^\{-1\})/\backslash log(t^\{-1\})\}$,