Abstract.A set X N of N points on the unit sphere is a spherical t-design if the average value of any polynomial of degree at most t over X N is equal to the average value of the polynomial over the sphere. This paper considers the characterization and computation of spherical t-designs on the unit sphere S 2 ⊂ R 3 when N ≥ (t + 1) 2 , the dimension of the space Pt of spherical polynomials of degree at most t. We show how to construct well conditioned spherical designs with N ≥ (t + 1) 2 points by maximizing the determinant of a matrix while satisfying a system of nonlinear constraints. Interval methods are then used to prove the existence of a true spherical t-design very close to the calculated points and to provide a guaranteed interval containing the determinant. The resulting spherical designs have good geometrical properties (separation and mesh norm). We discuss the usefulness of the points for both equal weight numerical integration and polynomial interpolation on the sphere and give an example.

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.

We propose a fully discrete hard thresholding polynomial approximation over a general region, named hard thresholding hyperinterpolation (HTH). This approximation is a weighted ℓ0regularized discrete least squares approximation under the same conditions of hyperinterpolation. Given an orthonormal basis of a polynomial space of total-degree not exceeding L and in view of exactness of a quadrature formula at degree 2L, HTH approximates the Fourier coefficients of a continuous function and obtains its coefficients by acting a hard thresholding operator on all approximated Fourier coefficients. HTH is an efficient tool to deal with noisy data because of the basis element selection ability. The main results of HTH for continuous and smooth functions are twofold: the L2 norm of HTH operator is bounded independently of the polynomial degree; and the L2 error bound of HTH is greater than that of hyperinterpolation but HTH performs well in denoising. We conclude with some numerical experiments to demonstrate the denoising ability of HTH over intervals, discs, spheres, spherical triangles and cubes.

This paper is concerned with the introduction of Tikhonov regularization into least squares approximation scheme on [−1, 1] by orthonormal polynomials, in order to handle noisy data. This scheme includes interpolation and hyperinterpolation as special cases. With Gauss quadrature points employed as nodes, coefficients of the approximation polynomial with respect to given basis are derived in an entry-wise closed form. Under interpolatory conditions, the solution to the regularized approximation problem is rewritten in forms of two kinds of barycentric interpolation formulae, by introducing only a multiplicative correction factor into both classical barycentric formulae. An L 2 error bound and a uniform error bound are derived, providing similar information that Tikhonov regularization is able to reduce the operator norm (Lebesgue constant) and the error term related to the level of noise, both by multiplying a correction factor which is less than one. Numerical examples show the benefits of Tikhonov regularization when data is noisy or data size is relatively small.

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