Nikol'skii Inequality for Algebraic Polynomials on a Multidimensional Euclidean Sphere / Arestov V. V.,Deikalova M. V. // PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS. - 2014. - V. 284, l. 1. - P. S9-S23.

ISSN/EISSN:

0081-5438 / 1531-8605

Type:

Article

Abstract:

We study the sharp Nikol'skii inequality between the uniform norm and the L-q norm of algebraic polynomials of a given (total) degree n >= 1 on the unit sphere Sm-1 of the Euclidean space R-m for 1 <= q < infinity. We prove that the polynomial rho(n) in one variable with unit leading coefficient that deviates least from zero in the space L-q(psi)(-1,1) of functions f such that |f|(q) is summable over (-1,1) with the Jacobi weight psi(t) - (1 - t)(alpha)(1 + t) (beta), alpha - ( m - 1)/2, beta - (m-3)/2, as a zonal polynomial in one variable t - epsilon(m), where x - (epsilon(1,),epsilon(2),...,epsilon(m)) epsilon Sm-1, is (in a certain sense, unique) extremal polynomial in the Nikol'skii inequality on the sphere Sm-1. The corresponding one-dimensional inequalities for algebraic polynomials on a closed interval are discussed.

Author keywords:

multidimensional Euclidean sphere; algebraic polynomials; Nikol'skii inequality; polynomials that deviate least from zero. TRIGONOMETRIC POLYNOMIALS; INTEGRAL-INEQUALITIES; DERIVATIVES; INTERVAL; METRICS

DOI:

10.1134/S0081543814020023

Web of Science ID:

ISI:000334277400002

Соавторы в МНС:

Другие поля

Поле	Значение
Month	APR
Publisher	MAIK NAUKA/INTERPERIODICA/SPRINGER
Address	233 SPRING ST, NEW YORK, NY 10013-1578 USA
Language	English
EISSN	1531-8605
Keywords-Plus	TRIGONOMETRIC POLYNOMIALS; INTEGRAL-INEQUALITIES; DERIVATIVES; INTERVAL; METRICS
Research-Areas	Mathematics
Web-of-Science-Categories	Mathematics, Applied; Mathematics
Author-Email	Vitalii.Arestov@urfu.ru Marina.Deikalova@urfu.ru
Funding-Acknowledgement	Russian Foundation for Basic Research {[}11-01-00462]; Ministry of Education and Science of the Russian Federation {[}1.1544.2011]; Program for State Support of Leading Universities of the Russian Federation {[}02.A03.21.0006]
Funding-Text	This work was supported by the Russian Foundation for Basic Research (project no. 11-01-00462), by the Ministry of Education and Science of the Russian Federation (state assignment no. 1.1544.2011), and by the Program for State Support of Leading Universities of the Russian Federation (agreement no. 02.A03.21.0006 of August 27, 2013).
Number-of-Cited-References	32
Usage-Count-Last-180-days	2
Usage-Count-Since-2013	15
Journal-ISO	Proc. Steklov Inst. Math.
Doc-Delivery-Number	AE8UG