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THE METRIC PROJECTIONS ONTO CLOSED CONVEX CONES IN A HILBERT SPACE

Part of: General convexity Normed linear spaces and Banach spaces; Banach lattices Approximations and expansions

Published online by Cambridge University Press:  11 February 2021

Yanqi Qiu Open the ORCID record for Yanqi Qiu [Opens in a new window]  and
Zipeng Wang
Yanqi Qiu
Affiliation:
Institute of Mathematics and Hua Loo-Keng Key Laboratory of Mathematics, AMSS, Chinese Academy of Sciences, Beijing100190, China. (yanqi.qiu@amss.ac.cn, yanqi.qiu@hotmail.com)
Zipeng Wang
Affiliation:
College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, P.R.China (zipengwang2012@gmail.com, zipengwang@cqu.edu.cn)
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Abstract

We study the metric projection onto the closed convex cone in a real Hilbert space $\mathscr {H}$ generated by a sequence $\mathcal {V} = \{v_n\}_{n=0}^\infty $ . The first main result of this article provides a sufficient condition under which the closed convex cone generated by $\mathcal {V}$ coincides with the following set:

$$ \begin{align*} \mathcal{C}[[\mathcal{V}]]: = \bigg\{\sum_{n=0}^\infty a_n v_n\Big|a_n\geq 0,\text{ the series }\sum_{n=0}^\infty a_n v_n\text{ converges in } \mathscr{H}\bigg\}. \end{align*} $$
Then, by adapting classical results on general convex cones, we give a useful description of the metric projection onto $\mathcal {C}[[\mathcal {V}]]$ . As an application, we obtain the best approximations of many concrete functions in $L^2([-1,1])$ by polynomials with nonnegative coefficients.

Keywords

closed convex conesmetric projectionsbest approximationpolynomials with nonnegative coefficients

MSC classification

Primary: 52A27: Approximation by convex sets
Secondary: 41A10: Approximation by polynomials 46C05: Hilbert and pre-Hilbert spaces
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 5 , September 2022 , pp. 1617 - 1650
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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