Weak Generalized Localization for Multiple Fourier Series Whose Rectangular Partial Sums Are Considered with Respect to Some Subsequence

I. L. Bloshanskii ; O. V. Lifantseva

Matematičeskie zametki, Tome 84 (2008) no. 3, pp. 334-347

Résumé

In this paper, we obtain the structural and geometric characteristics of some subsets of $\mathbb{T}^N=[-\pi,\pi]^N$ (of positive measure), on which, for the classes $L_p(\mathbb{T}^N)$, $p>1$, where $N\ge 3$, weak generalized localization for multiple trigonometric Fourier series is valid almost everywhere, provided that the rectangular partial sums $S_n(x;f)$  ($x\in\mathbb{T}^N$, $f\in L_p$) of these series have a “number” $n=(n_1,\dots,n_N)\in\mathbb Z_{+}^{N}$ such that some components $n_j$ are elements of lacunary sequences. For $N=3$, similar studies are carried out for generalized localization almost everywhere.

Détail

Keywords: multiple Fourier series, weak generalized localization, generalized localization, partial sum, lacunary sequence, Hölder's inequality, Orlicz class.