The radial part of a class of Sobolev polynomials on the unit ball
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The radial part of a class of Sobolev polynomials on the unit ball ´ ´ 2 · Miguel A. Pinar ˜ 2 Fatima Lizarte1 · Teresa E. Perez Received: 15 November 2019 / Accepted: 27 August 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this work, a Sobolev inner product on the unit ball of Rd involving the outward normal derivative is considered. A basis of mutually orthogonal polynomials associated with this inner product is constructed in terms of spherical harmonics and a radial part obtained from a family of univariate polynomials orthogonal with respect to a Sobolev inner product. The properties of this family constitute our main subject of study. In particular, we deduce algebraic properties, connection formulas, and some asymptotic properties. Finally, we show some numerical tests to illustrate the behavior of the roots of these univariate non-standard orthogonal polynomials. Keywords Orthogonal polynomials on the ball · Sobolev inner products · Normal derivatives Mathematics Subject Classification (2010) Primary: 33C50 · 42C05
1 Introduction For many years, univariate Sobolev orthogonal polynomials have been the main subject of a large number of research papers. The first motivation for the study of this Miguel A. Pi˜nar
[email protected] F´atima Lizarte [email protected] Teresa E. P´erez [email protected] 1
Departamento de Matem´aticas, Estad´ıstica y Computaci´on, Universidad de Cantabria, 39005, Santander, Spain
2
Departamento de Matem´atica Aplicada & Instituto de Matem´aticas (IEMath – GR), Universidad de Granada, 18071, Granada, Spain
Numerical Algorithms
kind of polynomials was the possibility of simultaneously approximate a function and its derivatives. A good survey on this subject can be found in [9]. Secondly, Sobolev orthogonal polynomials in several variables have a shorter history. Very often, the inner product considered is some modification of the classical inner product on Bd , the unit ball of Rd , involving an extra term containing the usual multivariate derivation operators (gradient, divergence, Laplacian, . . . ). The extension of Sobolev polynomials to the multivariate case started in [15], where the author studied a Sobolev inner product motivated by an application in the numerical solution of the nonlinear Poisson equation on the unit disk with zero boundary conditions. The central symmetry of the inner product plays an essential role in the construction of a basis of mutually orthogonal polynomials, which can be expressed in terms of spherical harmonics and a radial part given by univariate Sobolev orthogonal polynomials connected to Jacobi weights with varying parameters depending on its degree. In this direction, in [16] the author considers two different inner products involving the gradient operator on the ball. Using a similar reasoning, a family of explicit orthonormal basis is constructed for both inner products. In [11], an interesting result obtained by the authors is that orthogonal polynomials with respect to one of the Sobolev inne
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