Orthogonal Basis Function Over the Unit Circle with the Minimax Property

TL;DR

This paper introduces a new orthogonal basis for functions on the unit circle, combining sinusoidal azimuthal functions with radial polynomials designed to have equal amplitude extrema, inspired by Chebyshev polynomials.

Contribution

It presents a novel basis construction with a minimax property for functions on the unit circle, using numerical evaluation of overlap integrals involving generalized Fresnel integrals.

Findings

Basis functions exhibit equal amplitude minima and maxima along both coordinates.

Construction method involves numerical evaluation of overlap integrals.

Basis is suitable for applications requiring uniform amplitude properties.

Abstract

We construct an orthogonal basis of functions defined over the unit circle as the product of the common sinusoidal functions of the azimuth angle by radial functions which are essentially sines of a polynomials of the radial distance to the origin. The main impetus of this approach is to generate basis functions where the minima and maxima along both coordinates, the azimuth and the distance r to the center, have the same amplitude, akin to the Chebyshev polynomial basis of the one-dimensional unit interval. The construction is based on numerical evaluation of the overlap integrals, which have the format of generalized Fresnel integrals.