Nonuniform Sampling and Recovery of Multidimensional Bandlimited Functions by Gaussian Radial-basis Functions

Let (xn) ⊂ R d be a uniformly separated sequence which forms a Fourier frame for PWB2 , the space of square-integrable functions on R d whose Fourier transforms vanish outside the Euclidean unit ball B2. Given λ > 0 and f ∈ PWB2 , there is a unique sequence (aj) in l2 such that the function Iλ(f)(x) := X aje −λ‖x−xj‖ 2 2 , x∈R d , is continuous and square integrable on R, and satisfies the interpolatory conditions Iλ(f)(xn) = f(xn) for every n. It is shown that Iλ(f) converges to f in L2(R ), and also uniformly on R, as λ → 0.