Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-24T14:53:02.207Z Has data issue: false hasContentIssue false
  • English
  • Français

THE METRIC PROJECTIONS ONTO CLOSED CONVEX CONES IN A HILBERT SPACE

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

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. ([email protected], [email protected])
Zipeng Wang
Affiliation:
College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, P.R.China ([email protected], [email protected])
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Asimow, L. and Ellis, A. J., Convexity Theory and Its Applications in Functional Analysis, Vol. 16 of London Mathematical Society Monographs (Harcourt Brace Jovanovich, London, 1980).Google Scholar
Beck, A. and Hallak, N., On the convergence to stationary points of deterministic and randomised feasible descent directions methods, SIAM J. Optim. 30(1) (2020), 56–79.CrossRefGoogle Scholar
Borwein, J. M. and Lewis, A. S., Convex Analysis and Nonlinear Optimisation, Vol. 3 of CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, second edition (Springer, New York, 2006).CrossRefGoogle Scholar
Davis, C., Theory of positive linear dependence, Amer. J. Math. 76 (1954), 733–746.Google Scholar
Deutsch, F., Best approximation in Inner Product Spaces, Vol. 7 of CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC (Springer-Verlag, New York, 2001).Google Scholar
Deutsch, F., McCabe, J. H. and Phillips, G. M., Some algorithms for computing best approximations from convex cones, SIAM J. Numer. Anal. 12 (1975), 390–403.CrossRefGoogle Scholar
Domokos, A., Ingram, J. M. and Marsh, M. M., Projections onto closed convex sets in Hilbert spaces, Acta Math. Hungar. 152(1) (2017), 114–129.Google Scholar
Edwards, R. E., Functional Analysis. Theory and Applications. (Holt, Rinehart and Winston, New York, 1965).Google Scholar
Ingram, J. M. and Marsh, M. M., Projections onto convex cones in Hilbert space, J. Approx. Theory 64(3) (1991), 343–350.Google Scholar
Jurkat, W. B. and Lorentz, G. G., Uniform approximation by polynomials with positive coefficients, Duke Math. J. 28 (1961), 463–473.Google Scholar
McKinney, R. L., Positive bases for linear spaces, Trans. Amer. Math. Soc. 103 (1962), 131–148.Google Scholar
Nussbaum, R. D. and Walsh, B., Approximation by polynomials with nonnegative coefficients and the spectral theory of positive operators, Trans. Amer. Math. Soc. 350(6) (1998), 2367–2391.CrossRefGoogle Scholar
Stoer, J. and Witzgall, C., Convexity and Optimization in Finite Dimensions. I . Die Grundlehren der mathematischen Wissenschaften [Comprehensive Studies in Mathematics], Band 163 (Springer-Verlag, New York, 1970).Google Scholar
Toland, J. F., Self-adjoint operators and cones, J. London Math. Soc. (2) 53(1) (1996), 167–183.CrossRefGoogle Scholar
Wulbert, D. E., Continuity of metric projections, Trans. Amer. Math. Soc. 134 (1968), 335–341.Google Scholar
Zarantonello, E. H., Projections on convex sets in Hilbert space and spectral theory. I. Projections on convex sets, in Contributions to Nonlinear Functional Analysis (Proceedings of a Symposium Conducted by the Mathematics Research Center, University of Wisconsin–Madison, Academic Press, New York, 1971), pp. 237–341 (1971).CrossRefGoogle Scholar