Biorthonormal Systems in Freud-type Weighted Spaces with Infinitely Many Zeros - An Interpolation Problem

TL;DR

This paper constructs a complete and minimal biorthonormal system in Freud-type weighted spaces with infinitely many zeros, addressing an interpolation problem and analyzing convergence properties.

Contribution

It introduces a method to derive biorthonormal systems in weighted spaces with infinitely many zeros by modifying existing orthonormal systems, and explores associated interpolation and convergence issues.

Findings

Complete and minimal biorthonormal systems are constructed.

The paper solves an interpolation problem at infinitely many nodes.

Convergence properties of the solutions are analyzed.

Abstract

In a Freud-type weighted ($w$) space, introducing another weight ($v$) with infinitely many roots, we give a complete and minimal system with respect to $vw$, by deleting infinitely many elements from the original orthonormal system with respect to $w$. The construction of the conjugate system implies an interpolation problem at infinitely many nodes. Besides the existence, we give some convergence properties of the solution.