Hyperinterpolation on the square

Hyperinterpolation on the sphere

In this paper we survey hyperinterpolation on the sphere Sd, d ≥ 2. The hyperinterpolation operator Ln is a linear projection onto the space Pn(S) of spherical polynomials of degree≤ n, which is obtained from L2(S)-orthogonal projection onto Pn(S) by discretizing the integrals in the L2(S) inner products by a positive-weight numerical integration rule of polynomial degree of exactness 2n. Thus ...

On Generalized Hyperinterpolation on the Sphere

It is shown that second-order results can be attained by the generalized hyperinterpolation operators on the sphere, which gives an affirmative answer to a question raised by Reimer in Constr. Approx. 18(2002), no. 2, 183–203.

Hyperinterpolation at Xu Points and Interpolation at Padua Points in the Square: Computational Aspects

Hyperinterpolation in the cube

We construct an hyperinterpolation formula of degree n in the three-dimensional cube, by using the numerical cubature formula for the product Chebyshev measure given by the product of a (near) minimal formula in the square with Gauss-Chebyshev-Lobatto quadrature. The underlying function is sampled at N ∼ n/2 points, whereas the hyperinterpolation polynomial is determined by its (n + 1)(n + 2)(n...

The Uniform Error of Hyperinterpolation on the Sphere

This paper considers the problem of approximation of a continuous function on the unit sphere S ⊆ IR by a spherical polynomial from the space IPn of all spherical polynomials of degree ≤ n. For r = 3 it was shown in [16] that the hyperinterpolation approximation Lnf (which is the linear projection obtained by approximating the Fourier coefficients in the exact L2 orthogonal projection by a posi...