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This article is cited in 4 scientific papers (total in 4 papers)
Short communications
On the inequality of different metrics for multiple Fourier–Haar series
A. N. Bashirovaa, E. D. Nursultanovb a Faculty of Mechanics and Mathematics,
L.N. Gumilyov Eurasian National University,
13 Kazhymukan Munaitpasov St,
010008 Nur-Sultan, Kazakhstan
b M.V. Lomonosov Moscow State University,
Kazakhstan Branch,
11 Kazhymukan Munaitpasov St,
010010 Nur-Sultan, Kazakhstan
Abstract:
Let $1<p<q<\infty$, $f\in L_p[0, 1]$. Then, according to the inequality of different metrics due to S.M. Nikol'skii, for the sequence of norms of partial sums of the Fourier–Haar series $\{||S_{2^k}(f)||_{L_q}\}_{k=0}^\infty$ the following relation is true $||S_{2^k}(f)||_{L_q}=O\left(2^{k\left(\frac1p-\frac1q\right)}\right)$. In this paper, we study the asymptotic behavior of partial sums in the Lorentz spaces. In particular, it is obtained that $||S_{2^{k_1}2^{k_2}}(f)||_{L_{\overline{q}}}=o\left(2^{k_1\left(\frac1{p_1}-\frac1{q_1}\right)+k_2\left(\frac1{p_2}-\frac1{q_2}\right)}\right)$ for $f\in L_{\overline{p},\overline{\tau}}[0, 1]^2$.
Keywords and phrases:
Fourier series, Haar system, inequality of different metrics, anisotropic Lebesgue and Lorentz spaces.
Received: 01.08.2020
Citation:
A. N. Bashirova, E. D. Nursultanov, “On the inequality of different metrics for multiple Fourier–Haar series”, Eurasian Math. J., 12:3 (2021), 90–93
Linking options:
https://www.mathnet.ru/eng/emj417 https://www.mathnet.ru/eng/emj/v12/i3/p90
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