  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.
