# Further Spectral Properties of the Weighted Finite Transform Operator and Approximation in Weighted Sobolev Spaces

## TL;DR

This paper investigates the spectral properties of the weighted finite Fourier transform operator, focusing on the decay of singular values, bounds of GPSWFs, and their approximation capabilities in weighted Sobolev spaces, supported by numerical examples.

## Contribution

It provides new insights into the decay rate of singular values and approximation quality of GPSWFs in weighted Sobolev spaces, with detailed bounds and numerical illustrations.

## Key findings

- Singular values decay super-exponentially.
- GPSWFs have specific local bounds.
- Effective approximation in weighted Sobolev spaces.

## Abstract

In this work, we first give some mathematical preliminairies concerning the generelized prolate spheroidal wave functions(GPSWFs). This set of special functions have been introduced in [21]and [13] and they are defined as the infinite and countable set of the eigenfunctions of a weighted finite Fourier transform operator. Then, we show that the set of the singular values of this operator has a super-exponential decay rate. We also give some local estimates and bounds of these GPSWFs. As an application of the spectral properties of the GPSWFs and their associated eigenvalues, we give their quality of approximation in a weighted Sobolev space.Finally, we provide the reader with some numerical examples that illustrate the different results of this work.