论文标题
在希尔伯特空间中的封闭凸锥上的度量预测
The metric projections onto closed convex cones in a Hilbert space
论文作者
论文摘要
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} $中封闭凸锥上的度量投影。本文的第一个主要结果提供了足够的条件，在该条件下，我们可以识别由$ \ Mathcal {v} $生成的封闭凸锥，以及以下集：\ [\ [\ Mathcal {C} [[\ Mathcal {V}]：= = \ bigG \ sum_ { 0，\ text {系列} \ sum_ {n = 0}^\ infty a_n v_n \ text {comlem $ \ mathscr {h} $} $} \ bigg \}。 \]然后，通过对一般凸锥进行经典结果调整，我们对向量投影到$ \ Mathcal {c} [[\ Mathcal {v}}] $的度量投影有用。作为应用程序，我们通过具有非负系数的多项式获得了$ l^2（[ - 1,1]）$中许多混凝土功能的最佳近似值。
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 paper provides a sufficient condition under which we can identify the closed convex cone generated by $\mathcal{V}$ with the following set: \[ \mathcal{C}[[\mathcal{V}]]: = \bigg\{\sum_{n=0}^\infty a_n v_n\Big|a_n\geq 0,\text{ the series }\sum_{n=0}^\infty a_n v_n\text{ converges in $\mathscr{H}$}\bigg\}. \] Then, by adapting classical results on general convex cones, we give a useful description of the metric projection of a vector onto $\mathcal{C}[[\mathcal{V}]]$. As applications, we obtain the best approximations of many concrete functions in $L^2([-1,1])$ by polynomials with non-negative coefficients.
