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Title: On Bernstein inequalities for multivariate trigonometric polynomials in $L_{p}$, $0\leq p\leq \infty $ (English)
Author: Zhu, Laiyi
Author: Zhao, Xingjun
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 72
Issue: 2
Year: 2022
Pages: 449-459
Summary lang: English
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Category: math
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Summary: Let ${\mathbb T}_n$ be the space of all trigonometric polynomials of degree not greater than $n$ with complex coefficients. Arestov extended the result of Bernstein and others and proved that $ \| (1/n) T'_n \|_{p} \leq \| T_n \|_{p}$ for $0 \leq p \leq \infty $ and $T_n \in {\mathbb T}_n$. We derive the multivariate version of the result of Golitschek and Lorentz $$ \Bigl \| \Bigl | T_n \cos \alpha + \frac {1}{n} \nabla T_n \sin \alpha \Bigr |_{l_{\infty }^{(m)}} \Bigr \|_{p} \leq \| T_n \|_{p}, \quad 0 \leq p \leq \infty $$ for all trigonometric polynomials (with complex coeffcients) in $m$ variables of degree at most $n$. (English)
Keyword: univariate trigonometric polynomial
Keyword: multivariate trigonometric polynomial
Keyword: multivariate algebraic polynomial
Keyword: Bernstein inequality
Keyword: $L_{p}$-norm
MSC: 41A10
MSC: 41A17
idZBL: Zbl 07547214
idMR: MR4412769
DOI: 10.21136/CMJ.2021.0531-20
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Date available: 2022-04-21T19:02:15Z
Last updated: 2024-07-01
Stable URL: http://hdl.handle.net/10338.dmlcz/150411
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Reference: [1] Arestov, V. V.: On integral inequalities for trigonometric polynomials and their derivatives.Math. USSR, Izv. 18 (1982), 1-18 translation from Izv. Akad. Nauk SSSR, Ser. Mat. 45 1981 3-22. Zbl 0538.42001, MR 607574, 10.1070/IM1982v018n01ABEH001375
Reference: [2] Conway, J. B.: Functions of One Complex Variable II.Graduate Texts in Mathematics 159. Springer, New York (1995). Zbl 0887.30003, MR 1344449, 10.1007/978-1-4612-0817-4
Reference: [3] Golitschek, M. V., Lorentz, G. G.: Bernstein inequalities in $L_p$, $0 \leq p \leq \infty$.Rocky Mt. J. Math. 19 (1989), 145-156. Zbl 0738.42003, MR 1016168, 10.1216/RMJ-1989-19-1-145
Reference: [4] Rahman, Q. I., Schmeisser, G.: Les inégalités de Markoff et de Bernstein.Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics] 86. Les Presses de l'Université de Montréal, Montréal (1983), French. Zbl 0525.30001, MR 0729316
Reference: [5] Tung, S. H.: Bernstein's theorem for the polydisc.Proc. Am. Math. Soc. 85 (1982), 73-76. Zbl 0502.32004, MR 647901, 10.1090/S0002-9939-1982-0647901-6
Reference: [6] Zygmund, A.: A remark on conjugate series.Proc. Lond. Math. Soc., II. Ser. 34 (1932), 392-400. Zbl 0005.35301, MR 1576159, 10.1112/plms/s2-34.1.392
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