, Volume 42, Issue 1-2, pp 117-137,
Open Access This content is freely available online to anyone, anywhere at any time.
Date: 16 Nov 2012

How many Fourier samples are needed for real function reconstruction?

Abstract

In this paper we present some new results on the reconstruction of structured functions by a small number of equidistantly distributed Fourier samples. In particular, we show that real spline functions of order m with non-uniform knots containing N terms can be uniquely reconstructed by only m+N Fourier samples. Further, linear combinations of N non-equispaced shifts of a known low-pass function Φ can be reconstructed by N+1 Fourier samples. In the bivariate case, we consider the problem of function recovering by a small amount of Fourier samples on different lines through the origin. Our methods are based on the Prony method. The proofs given in this paper are constructive. Some numerical examples show the applicability of the proposed approach.