Hard thresholding hyperinterpolation over general regions

Abstract

This paper proposes a novel variant of hyperinterpolation, called hard thresholding hyperinterpolation. This approximation scheme of degree $n$ leverages a hard thresholding operator to filter all hyperinterpolation coefficients which approximate the Fourier coefficients of a continuous function by a quadrature rule with algebraic exactness $2n$. We prove that hard thresholding hyperinterpolation is the unique solution to an $\ell_0$-regularized weighted discrete least squares approximation problem. Hard thresholding hyperinterpolation is not only idempotent and commutative with hyperinterpolation, but also satisfies the Pythagorean theorem. By estimating the reciprocal of the Christoffel function, we demonstrate that the upper bound of the uniform norm of hard thresholding hyperinterpolation operator is not greater than that of hyperinterpolation operator. Hard thresholding hyperinterpolation possesses denoising and basis selection abilities as Lasso hyperinterpolation. To judge the denoising effects of hard thresholding and Lasso hyperinterpolations, this paper yields a criterion that combines the regularization parameter and the product of noise coefficients and signs of hyperinterpolation coefficients. Numerical examples on the spherical triangle and the cube demonstrate the denoising performance of hard thresholding hyperinterpolation.Comment: 19 pages, 7 figure