Hard thresholding hyperinterpolation over general regions
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We propose a fully discrete hard thresholding polynomial approximation over a general region, named hard thresholding hyperinterpolation (HTH). This approximation is a weighted ℓ_0-regularized 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 L_2 norm of HTH operator is bounded independently of the polynomial degree; and the L_2 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.
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