  We present an $\ell^2_2+\ell_1$-regularized discrete least squares
approximation over general regions under assumptions of hyperinterpolation,
named hybrid hyperinterpolation. Hybrid hyperinterpolation, using a soft
thresholding operator and a filter function to shrink the Fourier coefficients
approximated by a high-order quadrature rule of a given continuous function
with respect to some orthonormal basis, is a combination of Lasso and filtered
hyperinterpolations. Hybrid hyperinterpolation inherits features of them to
deal with noisy data once the regularization parameter and the filter function
are chosen well. We derive $L_2$ errors in theoretical analysis for hybrid
hyperinterpolation to approximate continuous functions with noise data on
sampling points. Numerical examples illustrate the theoretical results and show
that well chosen regularization parameters can enhance the approximation
quality over the unit-sphere and the union of disks.
