一个定义在半无穷区间上的希尔伯特空间及其基定理的函数论证明

On a Hilbert Space Defined on a Semi-Infinite Interval with a Functional Theoretic Proof of a Relating Basis Theorem

摘要: 定义在半无穷区间$\mathcal{Z}=\left[\frac{1}{4},+\infty \right)$ 上的有理函数系($\left\{g_{2n}:n\in N\right\}$ ，其中N表示非负整数全体)，给出了区间$\mathcal{Z}$ 上的希尔伯特空间($L^2\left(\mathcal{Z},w\right)$ )是研究者最新提出的研究内容。该空间中的基定理证明了$\left\{g_{2n}:n\in N\right\}$ 是$L^2\left(\mathcal{Z},w\right)$ 空间中的一组基。其定理的证明采用了等距同构的方法，未详细展开。本文首先给出$L^2\left(\mathcal{Z},w\right)$ 空间中的另一个刻画，并利用此刻画，采用函数论的方法，给出基定理的一个新证明，以展现此空间中的函数逼近结构。

Abstract: A family of rational functions $\left\{g_{2n}:n\in N\right\}$ (where N denotes the set of all nonnegative integers) defined on a semi-infinite interval $\mathcal{Z}=\left[\frac{1}{4},+\infty \right)$ , and a Hilbert space ($L^2\left(\mathcal{Z},w\right)$ ) on $\mathcal{Z}$ is a recently proposed research subject. $\left\{g_{2n}:n\in N\right\}$ was proved to be an orthonormal basis for $L^2\left(\mathcal{Z},w\right)$ in a theorem (the basis theorem) by using an isometry. The original proof is so brief that it might not have shown the hierarchy of function approximation relations clear enough. In this paper we give another characterization and raise examples of functions of several kinds in it. By taking advantages of this characterization, and by applying a function theory method, we offer a new proof for the basis theorem. We wish our deduction could show the hierarchical structurer of the function approximation in this space.