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Weak generalized localization for multiple Fourier series whose rectangular partial sums are considered with respect to some subsequence

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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 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.

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Authors and Affiliations

  1. Moscow State Regional University, Moscow, Russia

    I. L. Bloshanskii & O. V. Lifantseva

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  1. I. L. Bloshanskii
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  2. O. V. Lifantseva
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Correspondence to I. L. Bloshanskii.

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Original Russian Text © I. L. Bloshanskii, O. V. Lifantseva, 2008, published in Matematicheskie Zametki, 2008, Vol. 84, No. 3, pp. 334–347.

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Bloshanskii, I.L., Lifantseva, O.V. Weak generalized localization for multiple Fourier series whose rectangular partial sums are considered with respect to some subsequence. Math Notes 84, 314–327 (2008). https://doi.org/10.1134/S0001434608090022

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