Generalized sampling and the stable and accurate reconstruction of piecewise analytic functions from their Fourier coefficients

Abstract

Suppose that the first m m Fourier coefficients of a piecewise analytic function are given. Direct expansion in a Fourier series suffers from the Gibbs phenomenon and lacks uniform convergence. Nonetheless, in this paper we show that, under very broad conditions, it is always possible to recover an n n -term expansion in a different system of functions using only these coefficients. Such an expansion can be made arbitrarily close to the best possible n n -term expansion in the given system. Thus, if a piecewise polynomial basis is employed, for example, exponential convergence can be restored. The resulting method is linear, numerically stable and can be implemented efficiently in only O ( n m ) \mathcal {O} \left (n m\right ) operations. A key issue is how the parameter m m must scale in comparison to n n to ensure such recovery. We derive analytical estimates for this scaling for large classes of polynomial and piecewise polynomial bases. In particular, we show that in many important cases, including the case of piecewise Chebyshev polynomials, this scaling is quadratic: m = O ( n 2 ) m=\mathcal {O}\left (n^2\right ) . Therefore, with a system of polynomials that the user is essentially free to choose, one can restore exponential accuracy in n n and root-exponential accuracy in m m . This generalizes a result proved recently for piecewise Legendre polynomials.