New steps on Sobolev orthogonality in two variables

Abstract

MSC: 42C05 33C50Keywords: Orthogonal polynomials in two variables Sobolev orthogonal polynomials Classical orthogonal polynomials a b s t r a c t Sobolev orthogonal polynomials in two variables are defined via inner products involving gradients. Such a kind of inner product appears in connection with several physical and technical problems. Matrix second-order partial differential equations satisfied by Sobolev orthogonal polynomials are studied. In particular, we explore the connection between the coefficients of the second-order partial differential operator and the moment functionals defining the Sobolev inner product. Finally, some old and new examples are given.

On Sobolev orthogonal polynomials

“…The first case can be deduced by taking limit γ → −1 in the classical inner product with respect to ̟ α,β,γ , as observed in [18]. Identifying the correct form of the inner product is a major step.…”

Sobolev Orthogonal Polynomials on a Simplex

“…Instead we give an indication below on how the main terms of the inner products are identitified. We start from an observation in [6] that the monic orthogonal polynomials V (α,β,γ) k,n (x, y) are also orthogonal with respect to the inner product, when α, β, γ > −1,…”

“…which is a special case of Lemma 2.7. Since V α,β,γ k,n are well-defined if γ = −1, letting γ → −1 preserves the orthogonality, which is how the orthogonality in f, g α,β,−1 was established [6], but the method does not give the decomposition of the orthogonal space. Indeed, the second and the third terms in the right hand side of [f, g] α,β,γ make sense for γ = −1 as well, whereas for the first term we parametrize the integral over T 2 as…”

“…An observation in [6] shows that the monic basis for W α,β,γ is still a basis when γ = −1 and α, β > −1, and they are orthogonal with respect to an inner product that involves the first order derivatives. In contrast to one variable, there are only a handful papers on Sobolev orthogonal polynomials of several variables [6,10,12,13,17,18], see also [8]. It should be mentioned that the studies in [17,18] are motivated by problems in numerical solution of Poisson equations [5] and in optics.…”

“…In the case of d = 2, the simplex becomes a triangle and the weight function is usually denoted by W α,β,γ . An observation in [6] shows that the monic basis for W α,β,γ is still a basis when γ = −1 and α, β > −1, and they are orthogonal with respect to an inner product that involves the first order derivatives. In contrast to one variable, there are only a handful papers on Sobolev orthogonal polynomials of several variables [6,10,12,13,17,18], see also [8].…”