Let ϕ be a continuous function in L 2 (R) such that the sequence {ϕ(t − n)} n∈Z is a frame sequence in L 2 (R) and assume that the shift-invariant space V (ϕ) generated by ϕ has a multi-banded spectrum σ(V ). The main aim in this paper is to derive a multichannel sampling theory for the shift-invariant space V (ϕ). By using a type of Fourier duality between the spaces V (ϕ) and L 2 [0, 2π] we find necessary and sufficient conditions allowing us to obtain stable multi-channel sampling expansions in V (ϕ).Paley-Wiener space P W πσ of band-limited signals: In many common situations the available data are samples of some filtered versions of the signal itself. Following Ref. 19, there have been many generalizations and applications of the multi-channel sampling. See, for example, Refs. 6,8,18, 21 and 22. Although Shannon's sampling theory has had an enormous impact, it has a number of problems, as pointed out by Unser 20 : It relies on the use of ideal filters; the band-limited hypothesis is in contradiction with the idea of a finite duration signal; the band-limiting operation generates Gibbs oscillations; and finally, the sinc function has a very slow decay, which makes computation in the signal domain very inefficient.Moreover, many applied problems impose different a priori constraints on the type of functions. For this reason, sampling and reconstruction problems have been investigated in spline spaces, wavelet spaces, and general shift-invariant spaces. Indeed, in many practical applications, signals are assumed to belong to some shiftinvariant space of the form:. In most of cases in the mathematical literature, it is supposed that the sequence {ϕ(t − n)} n∈Z forms a Riesz basis for V (ϕ). See, for instance, 7,14,17,20, 23 and 24. Throughout this paper we assume that the sequence {ϕ(t − n)} n∈Z is a frame for V (ϕ) and that the spectrumOn the other hand, suppose that N linear time-invariant systems (filters) L j , j = 1, 2, . . . , N, are defined on the shift-invariant subspace V (ϕ) of L 2 (R). In mathematical terms we are dealing with continuous operators which commute with shifts. The recovery of any function f ∈ V (ϕ) from samples of the functions L j f , j = 1, 2, . . . , N, leads to a generalized sampling in V (ϕ).Our problem is the following: Given r, N positive integers and N real numbers 0 ≤ a j < r for 1 ≤ j ≤ N , find multi-channel sampling expansions like1) valid for any f ∈ V (ϕ), where the sequence of sampling functions {S j,n (t) : 1 ≤ j ≤ N, n ∈ Z} forms a frame or a Riesz basis for V (ϕ). Recently, García et al. 9-12 introduced a novel idea for developing a sampling theory on a shift-invariant space V (ϕ) by using an analogous of the Fourier duality between the spaces V (ϕ) and L 2 [0, 2π]. In particular, García and Pérez-Villalón 10 (see also Ref. 16) developed a multi-channel sampling procedure on a shift-invariant space V (ϕ), where ϕ is a continuous Riesz generator. Unlike the author's claim (see Sec. 4.1 in Ref. 10), the arguments used in Ref. 10 for the case of Riesz generator cannot be...