Constrained Orthogonal Polynomials

TL;DR

This paper introduces constrained orthogonal polynomials with zero average, generalizing classical polynomials, and explores their properties and applications in density functional theory and density fluctuations.

Contribution

It defines and analyzes a new class of orthogonal polynomials with a zero-average constraint, extending Laguerre and Legendre polynomials.

Findings

Explicit properties of constrained orthogonal polynomials are derived.

The structure of the subspace completing these polynomial sets is characterized.

A numerical example demonstrates their practical use.

Abstract

We define sets of orthogonal polynomials satisfying the additional constraint of a vanishing average. These are of interest, for example, for the study of the Hohenberg-Kohn functional for electronic or nucleonic densities and for the study of density fluctuations in centrifuges. We give explicit properties of such polynomial sets, generalizing Laguerre and Legendre polynomials. The nature of the dimension 1 subspace completing such sets is described. A numerical example illustrates the use of such polynomials.