THE METRIC PROJECTIONS ONTO CLOSED CONVEX CONES IN A HILBERT SPACE

We study the metric projection onto closed convex cone in a real Hilbert space $\mathscr{H}$ generated by sequence $\mathcal{V} = \{v_n\}_{n=0}^\infty$. The first main result of this paper provides sufficient condition under which we can identify $\mathcal{V}$ with following set: \[ \mathcal{C}[[\mathcal{V}]]: \bigg\{\sum_{n=0}^\infty a_n v_n\Big|a_n\geq 0,\text{ series }\sum_{n=0}^\infty v_n\text{ converges $\mathscr{H}$}\bigg\}. \] Then, adapting classical results on general cones, give useful description vector $\mathcal{C}[[\mathcal{V}]]$. As applications, obtain best approximations many concrete functions $L^2([-1,1])$ polynomials non-negative coefficients.