Sobolev orthogonal polynomials of high order in two variables defined on product domains

On Sobolev orthogonal polynomials on a triangle

“…Type II Sobolev orthogonal polynomials. For σ ∈ S 3 , let us define the univariate Sobolev inner product (7) (f,…”

“…Lemma 5.2. Let {q n,σ 0 (t)} n 0 and {q n,σ 1 (t)} n 0 be the sequences of orthogonal polynomials associated with the bilinear form defined in (7) with σ 0 and σ 1 . Then,…”

“…However, the study of Sobolev orthogonal polynomials in several variables is most recent. Some references include studies on the unit ball and the unit sphere [3-5, 11, 12, 15, 16, 18, 19], the simplex [1,20], product domains [6,7,9] and other interesting domains [14]. We remark that some Sobolev orthogonal polynomials are eigenfunctions of second order linear ordinary differential equations (see, for instance, [10] and the references therein) and second order linear partial differential equations [2].…”

Sobolev orthogonality of polynomial solutions of second-order partial differential equations

“…The tools and techniques for studying these multivariate polynomials are even fewer than in the univariate case. Some references include studies on the unit ball and the unit sphere (Dai and Xu 2011;Delgado et al 2016Delgado et al , 2013Li and Xu 2014;Lizarte et al 2021;Pérez et al 2013;Piñar and Xu 2009;Xu 2006Xu , 2008, the simplex (Atkas and Xu 2013;Xu 2017), and product domains (Dueñas et al 2021(Dueñas et al , 2017Fernández et al 2015). We remark that most of the results involve bilinear forms with first-order derivatives (Bracciali et al 2010;Marriaga et al 2021).…”