THE METRIC PROJECTIONS ONTO CLOSED CONVEX CONES IN A HILBERT SPACE. (11th September 2022) - Qiu, Yanqi; Wang, Zipeng
Abstract: We study the metric projection onto the closed convex cone in a real Hilbert space H generated by a sequence V = {v_n}_n=0^∞ . The first main result of this article provides a sufficient condition under which the closed convex cone generated by V coincides with the following set: align* C[[V]]: = {_n=0^∞ a_n v_n|a_n≥ 0, the series _n=0^∞ a_n v_n converges in H}. align* Then, by adapting classical results on general convex cones, we give a useful description of the metric projection onto C[[V]] . As an application, we obtain the best approximations of many concrete functions in L^2([-1, 1]) by polynomials with nonnegative coefficients.