Recovery Thresholding Hyperinterpolations in Signal Processing

Abstract: This paper introduces recovery thresholding hyperinterpolations, a novel class of methods for sparse signal reconstruction in the presence of noise. We develop a framework that integrates thresholding operators--including hard thresholding, springback, and Newton thresholding--directly into the hyperinterpolation structure to maintain sparsity during signal recovery. Our approach leverages Newton's method to minimize one-dimensional nonconvex functions, which we then extend to solve multivariable nonconvex regularization problems. The proposed methods demonstrate robust performance in reconstructing signals corrupted by both Gaussian and impulse noise. Through numerical experiments, we validate the effectiveness of these recovery thresholding hyperinterpolations for signal reconstruction and function denoising applications, showing their advantages over traditional approaches in preserving signal sparsity while achieving accurate recovery.