标题： 正交有理函数在半无限区间上 显示英文标题

标题： Orthonormal rational functions on a semi-infinite interval

摘要： 在本文中，我们提出了一种在半无限区间上的加权正交有理函数的新族。 我们以多种形式写出一系列整系数多项式，并推导出它们对应的微分方程。 这些方程不构成Sturm-Liouville问题。 我们通过乘以一些因子来克服这一缺点，从而得到一个无理函数序列。 我们推导出这个无理函数序列的各种生成函数，并找到其相关的Sturm-Liouville问题，这带来了正交性。 然后我们研究了一个定义在半无限区间上的函数的Hilbert空间，其内积由上述Sturm-Liouville问题确定的权函数所诱导。 我们列出了两个基。 一个是上述无理函数序列的偶数子序列，另一个是非正整数次幂函数。 我们提出了傅里叶级数展开的一个例子和一个插值的例子作为应用。 显示英文摘要

摘要： In this paper we propose a novel family of weighted orthonormal rational functions on a semi-infinite interval. We write a sequence of integer-coefficient polynomials in several forms and derive their corresponding differential equations. These equations do not form Sturm-Liouville problems. We overcome this disadvantage by multiplying some factors, resulting in a sequence of irrational functions. We deduce various generating functions of this sequence of irrational functions and find its associated Sturm-Liouville problems, which brings orthogonality. Then we study a Hilbert space of functions defined on a semi-infinite interval with its inner product induced by a weight function determined by the Sturm-Liouville problems mentioned above. We list two bases. One is the even subsequence of the irrational function sequence above and another one is the non-positive integer power functions. We raise one example of Fourier series expansion and one example of interpolation as applications.