Sufficient conditions for convergence of multiple Fourier series with Jk-lacunary sequence of rectangular partial sums in terms of Weyl multipliers
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 ) ∈ Z N , N ≥ 3, in which k...
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| Dokumentumtípus: | Cikk |
| Megjelent: |
Bolyai Institute, University of Szeged
Szeged
2017
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| Sorozat: | Acta scientiarum mathematicarum
83 No. 3-4 |
| Kulcsszavak: | Trigonometria, Matematika |
| Tárgyszavak: | |
| doi: | 10.14232/actasm-017-275-8 |
| Online Access: | http://acta.bibl.u-szeged.hu/50049 |
| Tartalmi kivonat: | 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 ) ∈ Z N , 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) = QN−k j=1 log(|ναj | + 2), where αj ∈ M \ Jk, ν = (ν1, . . . , νN ) ∈ Z N . 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 Orlicz 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. |
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| Terjedelem/Fizikai jellemzők: | 511-537 |
| ISSN: | 0001-6969 |