L2 error estimates for polynomial discrete penalized least-squares approximation on the sphere from noisy data

We consider discrete penalized least-squares approximation on the unit sphere Sd, where the minimizer is sought in the space PL(Sd) of spherical polynomials of degree ≤L. The penalized least-squares functional is the sum of two terms both involving the weights and the nodes of a positive weight quadrature rule with polynomial degree of exactness at least 2L. The first one is a discrete least-squares functional measuring the squared weighted ℓ2 discrepancy between the noisy data and the approximation. The second term is the product of a regularization parameter λ≥0 times a penalization term that can be interpreted as an approximation of a squared semi-norm in the Sobolev (Hilbert) space Hs(Sd). The approximation can be computed directly via a summation process and does not require the solving of a linear system. For λ=0 (which is only appropriate if there is almost no noise), our approximation becomes a case of hyperinterpolation. As λ>0 increases, less weight is given to data fitting and more weight is given to keeping the approximation smooth. We derive L2(Sd) error estimates for the approximation of functions from the Sobolev Hilbert space Hs(Sd), where s>d/2, from noisy data for the regularization parameter λ chosen (i) as λ=0 (hyperinterpolation), (ii) for general λ>0, and (iii) with Morozov’s discrepancy principle (an a posteriori parameter choice strategy). The L2(Sd) error estimates in case (i) and (iii) are in a sense order-optimal. Numerical experiments explore the approximation method and illustrate the theoretical results.