Fast Local Reconstruction Methods for Nonuniform Sampling in Shift-Invariant Spaces

Abstract. We present a new method for the fast reconstruction of a function f from its samples f(xj) under the assumption that f belongs to a shift invariant space V (φ). If the generator φ has compact support, then the reconstruction is local quite in contrast to methods based on band-limited functions. Using frame theoretic arguments, we show that the matrix of the corresponding linear system of equations is a positive-definite banded matrix. This special structure makes possible the fast local reconstruction algorithm in O(SJ) operations, where J is the number of samples and S is the support length of the generator φ. Further optimization can be achieved by means of data segmentation. Ample numerical simulation is provided.