2017/04/15 by I. L. Bloshanskii, Bloshanskii, I. L., S. K. Bloshanskaya +3
Mathematics · #42B05 (Secondary) #42C10 (Primary) #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #Mathematical Approximation and Integration
paper · pdf · doi:10.48550/arxiv.1704.04673
openalex publication_date 2017/04/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We obtain sufficient conditions for convergence (almost everywhere) of multiple trigonometric Fourier series of functions f in L2 in terms of Weyl multipliers. We consider the case where rectangular partial sums of Fourier series Sn(x;f) have indices n=(n1,…,nN) ∈ \mathbb ZN, N≥ 3, in which k (1≤ k≤ N-2) components on the places \j1,…,jk\=Jk ⊂ \1,…,N\ = M are elements of (single) lacunary sequences (i.e., we consider the, so called, multiple Fourier series with Jk-lacunary sequence of partial sums). We prove that for any sample Jk⊂ M the Weyl multiplier for convergence of these series has the form W(ν)=∏ j=1N-k log(|ναj|+2), where αj∈ M∖ Jk , ν=(ν1,…,νN)∈\mathbb ZN. So, the "one-dimensional" Weyl multiplier -- log(|⋅|+2) -- presents in W(ν) only on the places of "free" (nonlacunary) components of the vector ν. 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 Lp, 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 Sn(x;f) for N≥ 3 even in the class of continuous functions.