Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-04T18:04:14.427Z Has data issue: false hasContentIssue false
  • English
  • Français

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]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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. 

References

Cavaretta, A. S. and Hanchin, T., ‘A generalization of a result of G. Polya and its application to a continuous extension of the de la Vallée Poussin means’, J. Approx. Theory 158(2) (2009), 184–193.Google Scholar
Cheney, E. W., Hobby, C. R., Morris, P. D., Schurer, F. and Wulbert, D. E., ‘On the minimal property of the Fourier projection’, Trans. Amer. Math. Soc. 143 (1969), 249–258.Google Scholar
Cheney, E. W., Morris, P. D. and Price, K. H., ‘On an approximation operator of de la Vallée Poussin’, J. Approx. Theory 13 (1975), 375–391; collection of articles dedicated to G. G. Lorentz on the occasion of his sixty-fifth birthday, IV.Google Scholar
Horváth, Á., ‘Near-best approximation by a de la Vallée Poussin-type interpolatory operator’, Studia Sci. Math. Hungar. 49(1) (2012), 1–18.Google Scholar
Lambert, P. V., ‘On the minimum norm property of the Fourier projection in L 1-spaces’, Bull. Soc. Math. Belg. 21 (1969), 370–391.Google Scholar
Lewicki, G., Marino, G. and Pietramala, P., ‘Fourier-type minimal extensions in real L 1-spaces’, Rocky Mountain J. Math. 30(3) (2000), 1025–1037.CrossRefGoogle Scholar
Lewicki, G. and Micek, A., ‘Uniqueness of minimal Fourier-type extensions in L 1-spaces’, Monatsh. Math. 170(2) (2013), 161–178.Google Scholar
Lozinski, S. M., ‘On a class of linear operators’, Dokl. Akad. Nauk SSSR 61(2) (1948), 193–196.Google Scholar
Mehta, H., ‘The $L_{1}$ norm of the generalized de la Vallée Poussin kernel’, (2013), 1–12; arXiv:1311.1407 [math.CA].Google Scholar
Schurer, F. and Steutel, F. W., ‘Degree of approximation by the operators of de la Vallée Poussin’, Monatsh. Math. 87(1) (1979), 53–64.Google Scholar
Shekhtman, B. and Skrzypek, L., ‘Norming points and unique minimality of orthogonal projections’, Abstr. Appl. Anal. 2006 (2006), Article ID 42305, 17 pages.Google Scholar
Shekhtman, B. and Skrzypek, L., ‘On the uniqueness of the Fourier projection in L pspaces’, J. Concr. Appl. Math. 8(3) (2010), 439–447.Google Scholar
Singer, I., Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces (Springer, Berlin, 1970).Google Scholar
Telyakovskii, S. A., ‘Approximation to functions differentiable in Weyl’s sense by de la Vallée Poussin sums’, Dokl. Akad. Nauk SSSR 131 259–262; (in Russian), transl. Sov. Math. Dokl. 1 (1960), 204–243.Google Scholar
de la Vallée Poussin, C., ‘Sur la meilleure approximation des fonctions d’une variable réelle par des expressions d’ordre donné’, C. R. Acad. Sci. Paris 166 (1918), 799–802.Google Scholar