Fast and Stable Approximation of Analytic Functions from Equispaced Samples via Polynomial Frames

We consider approximating analytic functions on the interval [-1,1] from their values at a set of m+1 equispaced nodes. A result of Platte, Trefethen Kuijlaars states that fast and stable approximation from equispaced samples is generally impossible. In particular, any method that converges exponentially fast must also be exponentially ill-conditioned. We prove a positive counterpart to this ‘impossibility’ theorem. Our ‘possibility’ theorem shows that there is a well-conditioned method that provides exponential decay of the error down to a finite, but user-controlled tolerance ϵ > 0 , which in practice can be chosen close to machine epsilon. The method is known as polynomial frame approximation or polynomial extensions . It uses algebraic polynomials of degree n on an extended interval [-γ ,γ ] , γ > 1 , to construct an approximation on [-1,1] via a SVD-regularized least-squares fit. A key step in the proof of our main theorem is a new result on the maximal behaviour of a polynomial of degree n on [-1,1] that is simultaneously bounded by one at a set of m+1 equispaced nodes in [-1,1] and 1/ϵ on the extended interval [-γ ,γ ] . We show that linear oversampling, i.e. m = c n log (1/ϵ ) / √(γ ^2-1) , is sufficient for uniform boundedness of any such polynomial on [-1,1] . This result aside, we also prove an extended impossibility theorem, which shows that such a possibility theorem (and consequently the method of polynomial frame approximation) is essentially optimal.