@misc{10481/85368,
year = {2023},
month = {10},
url = {https://hdl.handle.net/10481/85368},
abstract = {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.},
organization = {Funding for open access publishing: Universidad de Granada/CBUA},
organization = {Funding for open access charge: Universidad de Granada},
publisher = {Springer Nature},
keywords = {Approximation on the ball},
keywords = {Inner product via Laplacians},
keywords = {Fourier expansions},
title = {Simultaneous Approximation via Laplacians on the Unit Ball},
doi = {10.1007/s00009-023-02509-9},
author = {Marriaga, Misael E. and Pérez Fernández, Teresa Encarnación and Recarte, Marlon J.},
}