On the Stability and Accuracy of Least Squares Approximations

We consider the problem of reconstructing an unknown function f on a domain X from samples of f at n randomly chosen points with respect to a given measure ρX . Given a sequence of linear spaces (Vm)m>0 with dim(Vm) = m ≤ n, we study the least squares approximations from the spaces Vm. It is well known that such approximations can be inaccurate when m is too close to n, even when the samples are noiseless. Our main result provides a criterion on m that describes the needed amount of regularization to ensure that the least squares method is stable and that its accuracy, measured in L(X, ρX), is comparable to the best approximation error of f by elements from Vm. We illustrate this criterion for various approximation schemes, such as trigonometric polynomials, with ρX being the uniform measure, and algebraic polynomials, with ρX being either the uniform or Chebyshev measure. For such examples we also prove similar stability results using deterministic samples that are equispaced with respect to these measures.