THE METRIC PROJECTIONS ONTO CLOSED CONVEX CONES IN A HILBERT SPACE. (11th September 2022)

• Record Type: Journal Article
• Authors: Qiu, Yanqi; Wang, Zipeng
• Is Part Of: Journal of the Institute of Mathematics of Jussieu. Volume 21:Number 5(2022)
• Page Start: 1617
• Page End: 1650
• Publication Date: 2022-09-11
• Subjects: closed convex cones -- metric projections -- best approximation -- polynomials with nonnegative coefficients; 52A27 -- 41A10 -- 46C05; Mathematics -- Periodicals; 510.5
• DOI: 10.1017/S1474748020000675
• Languages: English
• ISSNs: 1474-7480

Abstract

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.