Sampling and reconstruction in shift-invariant spaces on Rd

Let phi is an element of W (C, l(1)) such that {tau(n)phi : n is an element of Z(d)} forms a Riesz basis for V(phi). It is shown that Z(d) is a stable set of sampling for V(phi) if and only if Phi(+)(x) not equal 0, for every x is an element of T-d, where Phi(+)(x) := Sigma(n is an element of Zd) phi(n)e(2 pi in.x), x is an element of T-d. Sampling formulae are provided for reconstructing a function f is an element of V(phi) from uniform samples using Zak transform and complex analytic technique. The problem of sampling and reconstruction is discussed in the case of irregular samples also. The theory is illustrated with some examples, and numerical implementation for reconstruction of a function from its nonuniform samples is provided using MATLAB.