Simultaneous Approximation via Laplacians on the Unit Ball

We study the orthogonal structure on the unit ball Bd of Rd with respect to the Sobolev inner products f, g Δ = λL(f, g) + Bd Δ[(1 − x 2)f(x)] Δ[(1 − x 2)g(x)] dx, where L(f, g) = Sd−1 f(ξ) g(ξ) dσ(ξ) or L(f, g) = f(0)g(0), λ > 0, σ denotes the surface measure on the unit sphere Sd−1, and Δ is the usual Laplacian operator. Our main contribution consists in the study of orthogonal polynomials associated with ·, · Δ, giving their explicit expression in terms of the classical orthogonal polynomials on the unit ball, and proving that they satisfy a fourth-order partial differential equation, extending the well-known property for ball polynomials, since they satisfy a second-order PDE.We also study the approximation properties of the Fourier sums with respect to these orthogonal polynomials and, in particular, we estimate the error of simultaneous approximation of a function, its partial derivatives, and its Laplacian in the L2(Bd) space.Funding for open access publishing: Universidad de Granada/CBUAFunding for open access charge: Universidad de Granad

Approximation via gradients on the ball. The Zernike case

In this work, we deal in a d dimensional unit ball equipped with an inner product constructed by adding a mass point at zero to the classical ball inner product applied to the gradients of the functions. Apart from determining an explicit orthogonal polynomial basis, we study approximation properties of Fourier expansions in terms of this basis. In particular, we deduce relations between the partial Fourier sums in terms of the new orthogonal polynomials and the partial Fourier sums in terms of the classical ball polynomials. We also give an estimate of the approximation error by polynomials of degree at most n in the corresponding Sobolev space, proving that we can approximate a function by using its gradient. Numerical examples are given to illustrate the approximation behavior of the Sobolev basis.Ministerio de Ciencia, Innovacion y Universidades (MICINN), Spain PGC2018-096504-B-C33Comunidad de MadridUniversidad Rey Juan Carlos, Spain M2731FEDER/Junta de Andalucia, Spain A-FQM-246-UGR20MCIN/AEIFEDER, Spain funds PGC2018-094932-B-I00IMAG-Maria de Maeztu, Spain CEX2020-001105-

Bivariate Koornwinder–Sobolev Orthogonal Polynomials

Mathematics Subject Classification. 42C05, 33C50.The authors are grateful to the referee for his/her valuable comments and careful reading, which allowed us to improve this paper. The work of the first author (MEM) has been supported by Ministerio de Ciencia, Innovaci´on y Universidades (MICINN) grant PGC2018-096504-B-C33. Second and third authors (TEP and MAP) thank FEDER/Ministerio de Ciencia, Innovaci´on y Universidades—Agencia Estatal de Investigaci´on/PGC2018-094932-B-I00 and Research Group FQM-384 by Junta de Andaluc´ıa. This work is supported in part by the IMAG-Mar´ıa de Maeztu grant CEX2020-001105-M/AEI/10. 13039/501100011033.The so-called Koornwinder bivariate orthogonal polynomials are generated by means of a non-trivial procedure involving two families of univariate orthogonal polynomials and a function ρ(t) such that ρ(t)2 is a polynomial of degree less than or equal to 2. In this paper, we extend the Koornwinder method to the case when one of the univariate families is orthogonal with respect to a Sobolev inner product. Therefore, we study the new Sobolev bivariate families obtaining relations between the classical original Koornwinder polynomials and the Sobolev one, deducing recursive methods in order to compute the coefficients. The case when one of the univariate families is classical is analysed. Finally, some useful examples are given.Ministerio de Ciencia, Innovación y Universidades (MICINN) grant PGC2018-096504-B-C33EDER/Ministerio de Ciencia, Innovación y Universidades—Agencia Estatal de Investigación/PGC2018-094932-B-I00Research Group FQM-384 by Junta de AndalucíaIMAG-María de Maeztu grant CEX2020-001105-M/AEI/10.13039/501100011033Universidad de Granada/CBU