7 research outputs found
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