Lasso trigonometric polynomial approximation for periodic function recovery in equidistant points

Abstract

In this paper, we propose a fully discrete soft thresholding trigonometric polynomial approximation on $[-\pi,\pi],$ named Lasso trigonometric interpolation. This approximation is an $\ell_1$-regularized discrete least squares approximation under the same conditions of classical trigonometric interpolation on an equidistant grid. Lasso trigonometric interpolation is sparse and meanwhile it is an efficient tool to deal with noisy data. We theoretically analyze Lasso trigonometric interpolation for continuous periodic function. The principal results show that the $L_2$ error bound of Lasso trigonometric interpolation is less than that of classical trigonometric interpolation, which improved the robustness of trigonometric interpolation. This paper also presents numerical results on Lasso trigonometric interpolation on $[-\pi,\pi]$, with or without the presence of data errors.