On Bernstein inequalities for multivariate trigonometric polynomials in L p , 0 ≤ p ≤ ∞

Laiyi Zhu; Xingjun Zhao

Czechoslovak Mathematical Journal (2022)

  • Volume: 72, Issue: 2, page 449-459
  • ISSN: 0011-4642

Abstract

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Let 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 ≤ ∥ T n ∥ p for 0 ≤ p ≤ ∞ and T n ∈ 𝕋 n . We derive the multivariate version of the result of Golitschek and Lorentz T n cos α + 1 n ∇ T n sin α l ∞ ( m ) p ≤ ∥ T n ∥ p , 0 ≤ p ≤ ∞ for all trigonometric polynomials (with complex coeffcients) in m variables of degree at most n .

How to cite

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Zhu, Laiyi, and Zhao, Xingjun. "On Bernstein inequalities for multivariate trigonometric polynomials in $L_{p}$, $0\le p\le \infty $." Czechoslovak Mathematical Journal 72.2 (2022): 449-459. <http://eudml.org/doc/298302>.

@article{Zhu2022,
abstract = {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 $ \Vert (1/n) T^\{\prime \}_n \Vert _\{p\} \le \Vert T_n \Vert _\{p\}$ for $0 \le p \le \infty $ and $T_n \in \{\mathbb \{T\}\}_n$. We derive the multivariate version of the result of Golitschek and Lorentz \[ \Bigl \Vert \Bigl | T\_n \cos \alpha + \frac\{1\}\{n\} \nabla T\_n \sin \alpha \Bigr |\_\{l\_\{\infty \}^\{(m)\}\} \Bigr \Vert \_\{p\} \le \Vert T\_n \Vert \_\{p\}, \quad 0 \le p \le \infty \] for all trigonometric polynomials (with complex coeffcients) in $m$ variables of degree at most $n$.},
author = {Zhu, Laiyi, Zhao, Xingjun},
journal = {Czechoslovak Mathematical Journal},
keywords = {univariate trigonometric polynomial; multivariate trigonometric polynomial; multivariate algebraic polynomial; Bernstein inequality; $L_\{p\}$-norm},
language = {eng},
number = {2},
pages = {449-459},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On Bernstein inequalities for multivariate trigonometric polynomials in $L_\{p\}$, $0\le p\le \infty $},
url = {http://eudml.org/doc/298302},
volume = {72},
year = {2022},
}

TY - JOUR
AU - Zhu, Laiyi
AU - Zhao, Xingjun
TI - On Bernstein inequalities for multivariate trigonometric polynomials in $L_{p}$, $0\le p\le \infty $
JO - Czechoslovak Mathematical Journal
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 72
IS - 2
SP - 449
EP - 459
AB - 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 $ \Vert (1/n) T^{\prime }_n \Vert _{p} \le \Vert T_n \Vert _{p}$ for $0 \le p \le \infty $ and $T_n \in {\mathbb {T}}_n$. We derive the multivariate version of the result of Golitschek and Lorentz \[ \Bigl \Vert \Bigl | T_n \cos \alpha + \frac{1}{n} \nabla T_n \sin \alpha \Bigr |_{l_{\infty }^{(m)}} \Bigr \Vert _{p} \le \Vert T_n \Vert _{p}, \quad 0 \le p \le \infty \] for all trigonometric polynomials (with complex coeffcients) in $m$ variables of degree at most $n$.
LA - eng
KW - univariate trigonometric polynomial; multivariate trigonometric polynomial; multivariate algebraic polynomial; Bernstein inequality; $L_{p}$-norm
UR - http://eudml.org/doc/298302
ER -

References

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  1. Arestov, V. V., 10.1070/IM1982v018n01ABEH001375, Math. USSR, Izv. 18 (1982), 1-18 translation from Izv. Akad. Nauk SSSR, Ser. Mat. 45 1981 3-22. (1982) Zbl0538.42001MR607574DOI10.1070/IM1982v018n01ABEH001375
  2. Conway, J. B., 10.1007/978-1-4612-0817-4, Graduate Texts in Mathematics 159. Springer, New York (1995). (1995) Zbl0887.30003MR1344449DOI10.1007/978-1-4612-0817-4
  3. Golitschek, M. V., Lorentz, G. G., 10.1216/RMJ-1989-19-1-145, Rocky Mt. J. Math. 19 (1989), 145-156. (1989) Zbl0738.42003MR1016168DOI10.1216/RMJ-1989-19-1-145
  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. (1983) Zbl0525.30001MR0729316
  5. Tung, S. H., 10.1090/S0002-9939-1982-0647901-6, Proc. Am. Math. Soc. 85 (1982), 73-76. (1982) Zbl0502.32004MR647901DOI10.1090/S0002-9939-1982-0647901-6
  6. Zygmund, A., 10.1112/plms/s2-34.1.392, Proc. Lond. Math. Soc., II. Ser. 34 (1932), 392-400. (1932) Zbl0005.35301MR1576159DOI10.1112/plms/s2-34.1.392

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