THE METRIC PROJECTIONS ONTO CLOSED CONVEX CONES IN A HILBERT SPACE

• Published in: Journal of the Institute of Mathematics of Jussieu Vol. 21; no. 5; pp. 1617 - 1650
• Main Authors: Qiu, Yanqi, Wang, Zipeng
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.09.2022
• Subjects: Approximation; Cones; Euclidean space; Hilbert space; Mathematical analysis; Mathematics; Polynomials; 46C05; best approximation; closed convex cones; 41A10; metric projections; 52A27; polynomials with nonnegative coefficients
• ISSN: 1474-7480; 1475-3030
• DOI: 10.1017/S1474748020000675

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.