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.

我们研究了由序列 $\mathcal {V} = \{v_n\}_{n=0}^\infty $ 生成的真实希尔伯特空间 $\mathscr {H}$ 上封闭凸锥的度量投影。本文的第一个主要结果提供了一个充分条件，在该条件下由 $\mathcal {V}$ 生成的闭凸锥与以下集合重合： $$ \begin{align*} \mathcal{C}[[\mathcal{ V}]]: = \bigg\{\sum_{n=0}^\infty a_n v_n\Big|a_n\geq 0,\text{ 系列}\sum_{n=0}^\infty a_n v_n\text { 收敛于 } \mathscr{H}\bigg\}。\end{align*} $$ 然后，通过在一般凸锥上调整经典结果，我们给出了度量投影到 $\mathcal {C}[[\mathcal {V}]]$ 上的有用描述. 作为一个应用程序，我们通过具有非负系数的多项式获得了 $L^2([-1,1])$ 中许多具体函数的最佳近似值。