Simultaneous Approximation via Laplacians on the Unit Ball

We study the orthogonal structure on the unit ball B^d of ℝ^d with respect to the Sobolev inner products ⟨ f,g⟩ _Δ =λ ℒ(f,g) + ∫ _B^dΔ [(1-‖ x‖ ^2) f(x)] Δ [(1-‖ x‖ ^2) g(x)] dx, where ℒ(f,g) = ∫ _S^d-1f(ξ ) g(ξ ) dσ (ξ ) or ℒ(f,g) = f(0) g(0) , λ >0 , σ denotes the surface measure on the unit sphere S^d-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 L^2(B^d) space.