Regularized Least Squares Approximations on the Sphere Using Spherical Designs

Abstract

We consider polynomial approximation on the unit sphere S 2 = {(x, y, z) ∈ R 3 : x 2 + y 2 + z 2 = 1} by a class of regularized discrete least squares methods with novel choices for the regularization operator and the point sets of the discretization. We allow different kinds of rotationally invariant regularization operators, including the zero operator (in which case the approximation includes interpolation, quasi-interpolation, and hyperinterpolation); powers of the negative Laplace-Beltrami operator (which can be suitable when there are data errors); and regularization operators that yield filtered polynomial approximations. As node sets we use spherical t-designs, which are point sets on the sphere which when used as equal-weight quadrature rules integrate all spherical polynomials up to degree t exactly. More precisely, we use well conditioned spherical t-designs obtained in a previous paper by maximizing the determinants of the Gram matrices subject to the spherical design constraint. For t ≥ 2L and an approximating polynomial of degree L it turns out that there is no linear algebra problem to be solved and the approximation in some cases recovers known polynomial approximation schemes, including interpolation, hyperinterpolation, and filtered hyperinterpolation. For t ∈ [L, 2L) the linear system needs to be solved numerically. Finally, we give numerical examples to illustrate the theoretical results and show that well chosen regularization operator and well conditioned spherical t-designs can provide good polynomial approximation on the sphere, with or without the presence of data errors.

Spectral Numerical Exterior Calculus Methods for Differential Equations on Radial Manifolds

“…In hyperinterpolation, functions are oversampled to avoid many of the inherent issues associated with trying to design an optimal collection of nodes for Lagrange interpolation [25,29,30,36]. This allows for functions to be approximated through L 2 -orthogonal projections using exact quadratures up to a desired order [2,30]. To achieve discretizations with favorable symmetry on the sphere, we use the nodes of Lebedev quadrature [17,18].…”

“…We use hyperinterpolation to obtain a continuum approximation to fields on the manifold surface [2,30]. To obtain a continuum representation of a function f on the surface, we perform an L 2 -orthogonal projection P to the space spanned by spherical harmonics up to order L/2 ,…”

Corrigendum: Regularized Least Squares Approximations on the Sphere Using Spherical Designs

“…This note provides a corrigendum to Proposition 5.1 of [1] which gave an expression for the Lebesgue constant which is in fact an upper bound.…”

“…The formula (1.2) is proved by first showing that the right-hand side of (1.2) is an upper bound on the Lebesgue constant, as in [1], and then noting that there exists a function f * ∈ C(S 2 ) with f * C(S 2 ) = 1 such that…”

Uniform approximation on the sphere by least squares polynomials

“…Moreover, it is known that we can get uniformly bounded Lebesgue constants by filtered hyperinterpolation quasi-projections, under suitable assumptions on the filter coefficients which define the generalized de la Vallée Poussin mean [7,20,17]. In particular, for any 0 < θ < 1, if we set m = θn ( · being the floor function) and apply (1) with µ = 4n to the following arithmetic mean of Fourier sums…”