Sharp results on irregular sampling of bandlimited functions

Abstract

We discuss the reconstruction of bandlimited functions for randomly sampled values and give an algorithm that works provided the sampling density is above the Nyquist rate.

Let f be of finite energy and bandlimited with bandwidth 2ω, i.e., f ∊ L²(R), supp $\hat f \subseteq$ [−ω,ω], and let ... < x_i−1 < x_i < x_i+1 < ... be a random sampling set, such that its density δ:= supⁱ(x_i+1 − x_i) < π/ω, i.e., arbitrarily close to the Nyquist rate. If f_n is the result of the algorithm after n iterations, then the rate of convergence of f_n to the original function f is ∥f − f_n∥₂ ≤ (δω/π)_n+1(π +δω )(π − δω )₋₁ ∣∥f∥₂. This allows for good estimates of the number of iterations required to achieve a certain reconstruction accuracy.

In contrast to recent reconstruction methods: (1) an explicit and optimal estimate for the sampling density required for the convergence of the algorithm is derived, and (2) the algorithm functions independently of the sampling geometry—as long as the sampling density is higher than the Nyquist rate.