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Equiconvergence of Expansions in Multiple Fourier Series and in Fourier Integrals with “Lacunary Sequences of Partial Sums”
I. L. Bloshanskii, D. A. Grafov Moscow State Region University
Abstract:
We investigate the equiconvergence on $\mathbb T^N=[-\pi,\pi)^N$ of expansions in multiple trigonometric Fourier series and in the Fourier integrals of functions $f\in L_p({\mathbb T}^N)$ and $g\in L_p({\mathbb R}^N)$, $p>1$, $N\ge 3$, $g(x)=f(x)$ on $\mathbb T^N$, in the case where the “partial sums” of these expansions, i.e., $S_n(x;f)$ and $J_\alpha(x;g)$, respectively, have “numbers” $n\in {\mathbb Z}^N$ and $\alpha\in {\mathbb R}^N$ ($n_j=[\alpha_j]$, $j=1,\dots,N$, $[t]$ is the integral part of $t\in \mathbb R^1$) containing $N-1$ components which are elements of “lacunary sequences.”
Keywords:
multiple Fourier series, multiple Fourier integrals, convergence almost everywhere, lacunary sequence.
Received: 10.03.2014 Revised: 04.10.2014
Citation:
I. L. Bloshanskii, D. A. Grafov, “Equiconvergence of Expansions in Multiple Fourier Series and in Fourier Integrals with “Lacunary Sequences of Partial Sums””, Mat. Zametki, 99:2 (2016), 186–200; Math. Notes, 99:2 (2016), 196–209
Linking options:
https://www.mathnet.ru/eng/mzm10503https://doi.org/10.4213/mzm10503 https://www.mathnet.ru/eng/mzm/v99/i2/p186
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