Simultaneous Approximation via Laplacians on the Unit Ball Marriaga, Misael E. Pérez Fernández, Teresa Encarnación Recarte, Marlon J. Approximation on the ball Inner product via Laplacians Fourier expansions 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. 2023-10-31T11:01:23Z 2023-10-31T11:01:23Z 2023-10-13 info:eu-repo/semantics/article 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] https://hdl.handle.net/10481/85368 10.1007/s00009-023-02509-9 eng http://creativecommons.org/licenses/by/4.0/ info:eu-repo/semantics/openAccess Atribución 4.0 Internacional Springer Nature