  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.
