Orthonormal rational functions on a semi-infinite interval

Abstract

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.