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- W2605775038 abstract "We obtain sufficient conditions for convergence (almost everywhere) of multiple trigonometric Fourier series of functions $f$ in $L_2$ in terms of Weyl multipliers. We consider the case where rectangular partial sums of Fourier series $S_n(x;f)$ have indices $n=(n_1,dots,n_N) in mathbb Z^N$, $Nge 3$, in which $k$ $(1leq kleq N-2)$ components on the places ${j_1,dots,j_k}=J_k subset {1,dots,N} = M$ are elements of (single) lacunary sequences (i.e., we consider the, so called, multiple Fourier series with $J_k$-lacunary sequence of partial sums). We prove that for any sample $J_ksubset M$ the Weyl multiplier for convergence of these series has the form $W(nu)=prod limits_{j=1}^{N-k} log(|nu_{{alpha}_j}|+2)$, where $alpha_jin Msetminus J_k $, $nu=(nu_1,dots,nu_N)in{mathbb Z}^N$. So, the one-dimensional Weyl multiplier -- $log(|cdot|+2)$ -- presents in $W(nu)$ only on the places of free (nonlacunary) components of the vector $nu$. Earlier, in the case where $N-1$ components of the index $n$ are elements of lacunary sequences, convergence almost everywhere for multiple Fourier series was obtained in 1977 by M.Kojima in the classes $L_p$, $p>1$, and by D.K.Sanadze, Sh.V.Kheladze in Orlizc class. Note, that presence of two or more free components in the index $n$ (as follows from the results by Ch.Fefferman (1971)) does not guarantee the convergence almost everywhere of $S_n(x;f)$ for $Ngeq 3$ even in the class of continuous functions." @default.
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- W2605775038 date "2017-04-15" @default.
- W2605775038 modified "2023-10-16" @default.
- W2605775038 title "Sufficient conditions for convergence of multiple Fourier series with $J_k$-lacunary sequence of rectangular partial sums in terms of Weyl multipliers" @default.
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- W2605775038 doi "https://doi.org/10.48550/arxiv.1704.04673" @default.
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