This chapter is concerned with numerical integration over the unit sphere $${\mathbb
{S}}^{2}\subset {\mathbb {R}}^{3} $$. We first discuss basic facts about numerical integration …

Let φ k be an orthonormal system on a quasi-metric measure space X, ℓ k be a
nondecreasing sequence of numbers with lim k→∞ ℓ k=∞. A diffusion polynomial of degree …

Recently, fast and reliable algorithms for the evaluation of spherical harmonic expansions
have been developed. The corresponding sampling problem is the computation of Fourier …

This paper develops a fully discrete soft thresholding polynomial approximation over a
general region, named Lasso hyperinterpolation. This approximation is an \ell_1-regularized …

Let X be a compact, smooth, connected, Riemannian manifold without boundary, G: X× X→
R be a kernel. Analogous to a radial basis function network, an eignet is an expression of …

In industry and economics, the most common solutions of partial differential equations
involving multivariate numerical integration over cuboids include techniques of iterated one …

On the half line, we introduce a new sequence of near-best uniform approximation
polynomials, easily computable by the values of the approximated function at a truncated …

The objective of sampling in Euclidean spaces is to deal with the reconstruction of a
(bandlimited) function (or more generally by a function in an appropriate Hilbert space) by …

This paper considers a fully discrete filtered polynomial approximation on the unit sphere S^
d. For f ∈ C (S^ d), V_ L, N^(a)\, f is a polynomial approximation which is exact for all …

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 …