Article
| 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: | 2022-09-08 |
| 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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