Optimally space localized polynomials with applications in signal processing

TL;DR

This paper develops explicit formulas for polynomials optimally localized in space, useful for filtering peaks in periodic signals, with applications demonstrated in signal processing.

Contribution

It introduces a method to determine polynomials that are optimally localized in space, including explicit formulas and applications in signal filtering.

Findings

Explicit formulas for optimal space localized polynomials

Relation of mean value to uncertainty principle for Jacobi polynomials

Application of optimal polynomials as filters in signal processing

Abstract

For the filtering of peaks in periodic signals, we specify polynomial filters that are optimally localized in space. The space localization of functions having an expansion in terms of orthogonal polynomials is thereby measured by a generalized mean value. Solving an optimization problem including this mean value, we determine those polynomials out of a polynomial space that are optimally localized. We give explicit formulas for these optimally space localized polynomials and determine in the case of the Jacobi polynomials the relation of the generalized mean value to the position variance of a well-known uncertainty principle. Further, we will consider the Hermite polynomials as an example on how to get optimally space localized polynomials in a non-compact setting. Finally, we investigate how the obtained optimal polynomials can be applied as filters in signal processing.