Fast and stable approximation of analytic functions from equispaced samples via polynomial frames

Abstract

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 ǫ > 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 leastsquares 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/ǫ on the extended interval [−γ, γ]. We show that linear oversampling, i.e., m = cn log(1/ǫ)/ 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.

AAA interpolation of equispaced data

“…The fit is carried out by least-squares or regularized least-squares, often simply by means of the backslash operator in MATLAB, as has been analyzed by Adcock,Huybrechs,and Vaquero [4,30] and Lyon [36]. A related idea is polynomial extension, in which f is approximated by polynomials expressed in a basis of orthogonal polynomials defined on an interval [−T , T ] [5]. A third possibility is RBF extension, in which f is approximated by smooth RBFs whose centers extend outside [−1, 1] [26,39].…”

Efficient function approximation on general bounded domains using splines on a Cartesian grid

“…We do not consider other types of oversampling and observe that proving sufficient and necessary conditions for oversampling is currently an open problem in related approximation schemes including Fourier extension. Most recently, the sufficiency of linear oversampling was shown for polynomial extension frames, the first such result [4].…”

AAA interpolation of equispaced data

“…The fit is carried out by least-squares or regularized least-squares, often simply by means of the backslash operator in MATLAB, as has been analyzed by Adcock,Huybrechs,and Vaquero [4,28] and Lyon [34]. A related idea is polynomial extension, in which f is approximated by polynomials expressed in a basis of orthogonal polynomials defined on an interval [−T, T ] [5]. A third possibility is RBF extension, in which f is approximated by smooth RBFs whose centers extend outside [−1, 1] [24,37].…”