Sampling and recovery of bandlimited functions and applications to signal processing

摘要

Bandlimited functions, i.e square integrable functions on R~d, d ∈N, whose Fourier transforms have bounded support, are widely used to represent signals. One problem which arises, is to find stable recovery formulae, based on evaluations of these functions at given sample points. We start with the case of equally distributed sampling points and present a method of Daubechies and DeVore to approximate bandlimited functions by quantized data. In the case that the sampling points are not equally distributed this method will fail. We are suggesting to provide a solution to this problem in the case of scattered sample points by first approximating bandlimited functions using linear combinations of shifted Gaussians. In order to be able to do so we prove the following interpolation result. Let (x_j: j ∈ Z) ? R be a Rieszbasis sequence. For λ >0 and f ∈ PW, the space of square-integrable functions on R, whose Fourier transforms vanish outside of [-1, 1], there is a unique sequence (a_j) ∈ f_2 (Z), so that the function is continuous, square integrable, and satisfies the interpolatory conditions I_λ(f)(x_k) = f(x_k), for all k ∈ Z. It is shown that I_λ(f) converges to f in L_2(R~d) and uniformly on R, as λ →0~+.