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ON THE MINIMAL PROPERTY OF DE LA VALLÉE POUSSIN’S OPERATOR

Part of: General theory of linear operators Harmonic analysis in one variable

Published online by Cambridge University Press:  08 October 2014

BEATA DEREGOWSKA  and
BARBARA LEWANDOWSKA
BEATA DEREGOWSKA*
Affiliation:
Faculty of Mathematics and Computer Science, Jagiellonian University, Lojasiewicza 6, 30-048 Krakow, Poland email [email protected]
BARBARA LEWANDOWSKA
Affiliation:
Faculty of Mathematics and Computer Science, Jagiellonian University, Lojasiewicza 6, 30-048 Krakow, Poland email [email protected]
*
✉[email protected]
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Abstract

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Let $X={\mathcal{C}}_{0}(2{\it\pi})$ or $X=L_{1}[0,2{\it\pi}]$. Denote by ${\rm\Pi}_{n}$ the space of all trigonometric polynomials of degree less than or equal to $n$. The aim of this paper is to prove the minimality of the norm of de la Vallée Poussin’s operator in the set of generalised projections ${\mathcal{P}}_{{\rm\Pi}_{n}}(X,\,{\rm\Pi}_{2n-1})=\{P\in {\mathcal{L}}(X,{\rm\Pi}_{2n-1}):P|_{{\rm\Pi}_{n}}\equiv \text{id}\}$.

Keywords

de la Vallée Poussin’s operatorgeneralised projectionsDirichlet kernel

MSC classification

Primary: 42A10: Trigonometric approximation
Secondary: 47A58: Operator approximation theory
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 91 , Issue 1 , February 2015 , pp. 129 - 133
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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