On the possibility of fast stable approximation of analytic functions from equispaced samples via polynomial frames
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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 possibility 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 the possibility theorem (and consequently the method of polynomial frame approximation) is essentially optimal.
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