×

On Bernstein inequalities for multivariate trigonometric polynomials in \(L_p\), \(0\leq p\leq\infty\). (English) Zbl 07547214

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 \(n\).

MSC:

41A10 Approximation by polynomials
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Arestov, V. V., On integral inequalities for trigonometric polynomials and their derivatives, Math. USSR, Izv., 18, 1-18 (1982) · Zbl 0538.42001
[2] Conway, J. B., Functions of One Complex Variable II (1995), New York: Springer, New York · Zbl 0887.30003
[3] Golitschek, M. V.; Lorentz, G. G., Bernstein inequalities in Lp, 0 ⩽ p ⩽ ∞, Rocky Mt. J. Math., 19, 145-156 (1989) · Zbl 0738.42003
[4] Rahman, Q. I.; Schmeisser, G., Les inégalités de Markoff et de Bernstein (1983), Montréal: Les Presses de l’Université de Montréal, Montréal · Zbl 0525.30001
[5] Tung, S. H., Bernstein’s theorem for the polydisc, Proc. Am. Math. Soc., 85, 73-76 (1982) · Zbl 0502.32004
[6] Zygmund, A., A remark on conjugate series, Proc. Lond. Math. Soc., II. Ser., 34, 392-400 (1932) · Zbl 0005.35301
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.