Sharp integral inequalities for fractional derivatives of trigonometric polynomials / Arestov Vitalii V.,Glazyrina Polina Yu. // JOURNAL OF APPROXIMATION THEORY. - 2012. - V. 164, l. 11. - P. 1501-1512.

ISSN/EISSN:

0021-9045 / нет данных

Type:

Article

Abstract:

We study sharp estimates of integral functionals for operators on the set T-n of real trigonometric polynomials f(n) of degree n >= 1 in terms of the uniform norm parallel to f(n)parallel to(C2 pi) of the polynomials and similar questions for algebraic polynomials on the unit circle of the complex plane. P. Erdos, A.P. Calderon, G. Klein, L.V. Taikov, and others investigated such inequalities. In this paper, we, in particular, show that the sharp inequality parallel to D-alpha f(n)parallel to(q) <= n(alpha) parallel to cos t parallel to(q)parallel to f(n)parallel to(infinity) holds on the set T-n for the Weyl fractional derivatives D-alpha f(n) of order alpha >= 1 for 0 <= q < infinity. For q = infinity (alpha >= 1), this fact was proved by Lizorkin (1965) {[}12]. For 1 <= q < infinity and positive integer a, the inequality was proved by Taikov (1965) {[}23]; however, in this case, the inequality follows from results of an earlier paper by Calderon and Klein (1951) {[}6]. (C) 2012 Elsevier Inc. All rights reserved.

Author keywords:

Trigonometric polynomial; Algebraic polynomial; Derivative of fractional order; Bernstein inequality; Szego inequality

DOI:

10.1016/j.jat.2012.08.004

Web of Science ID:

ISI:000309894900004

Соавторы в МНС:

Другие поля

Поле	Значение
Month	NOV
Publisher	ACADEMIC PRESS INC ELSEVIER SCIENCE
Address	525 B ST, STE 1900, SAN DIEGO, CA 92101-4495 USA
Language	English
Research-Areas	Mathematics
Web-of-Science-Categories	Mathematics
Author-Email	vitalii.arestov@usu.ru polina.glazyrina@usu.ru
Funding-Acknowledgement	Russian Foundation for Basic Research {[}11-01-00462]; Ministry of Education and Science of the Russian Federation {[}1.1544.2011]; Ural Branch of the Russian Academy of Sciences {[}12-S-1-1018]; Siberian Branch of the Russian Academy of Sciences {[}12-S-1-1018]
Funding-Text	This work was supported by the Russian Foundation for Basic Research (project no. 11-01-00462) and by the Ministry of Education and Science of the Russian Federation within the state task to higher education institutions for fundamental and applied research (project no. 1.1544.2011). The work of the second author is also supported by the Integration Project for Fundamental Research of the Ural and Siberian Branches of the Russian Academy of Sciences (project no. 12-S-1-1018).
Number-of-Cited-References	28
Usage-Count-Last-180-days	1
Usage-Count-Since-2013	20
Journal-ISO	J. Approx. Theory
Doc-Delivery-Number	021OJ