Simultaneous Approximation via Laplacians on the Unit Ball
Metadatos
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Springer Nature
Materia
Approximation on the ball Inner product via Laplacians Fourier expansions
Date
2023-10-13Referencia bibliográfica
Marriaga, M.E., Pérez, T.E. & Recarte, M.J. Simultaneous Approximation via Laplacians on the Unit Ball. Mediterr. J. Math. 20, 316 (2023). [https://doi.org/10.1007/s00009-023-02509-9]
Patrocinador
Funding for open access publishing: Universidad de Granada/CBUA; Funding for open access charge: Universidad de GranadaRésumé
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.