Block orthogonal polynomials: I. Definition and properties

TL;DR

This paper introduces block orthogonal polynomials, defining their properties and a method for their construction, with potential applications to classical orthogonal polynomials in future work.

Contribution

It defines block orthogonal polynomials with respect to two scalar products and provides an explicit Gram-Schmidt procedure for their construction.

Findings

Defined block orthogonal polynomials with two scalar products

Provided a two-step Gram-Schmidt orthogonalization method

Analyzed properties and differences from standard orthogonal polynomials

Abstract

Constrained orthogonal polynomials have been recently introduced in the study of the Hohenberg-Kohn functional to provide basis functions satisfying particle number conservation for an expansion of the particle density. More generally, we define block orthogonal (BO) polynomials which are orthogonal, with respect to a first Euclidean scalar product, to a given $i$-dimensional subspace ${\cal E}_i$ of polynomials associated with the constraints. In addition, they are mutually orthogonal with respect to a second Euclidean scalar product. We recast the determination of these polynomials into a general problem of finding particular orthogonal bases in an Euclidean vector space endowed with distinct scalar products. An explicit two step Gram-Schmidt orthogonalization (G-SO) procedure to determine these bases is given. By definition, the standard block orthogonal (SBO) polynomials are associated with a choice of ${\cal E}_i$ equal to the subspace of polynomials of degree less than $i$. We investigate their properties, emphasizing similarities to and differences from the standard orthogonal polynomials. Applications to classical orthogonal polynomials will be given in forthcoming papers.