Generalized sampling in shift-invariant spaces with multiple stable generators

Abstract

The aim of this paper is to derive stable generalized sampling in a shift-invariant space with stable generators. This is done in the light of the theory of frames in the product Hilbert space L 2 (0, 1) := L 2 (0, 1) × · · · × L 2 (0, 1) ( times). The generalized samples are expressed as the frame coefficients of an appropriate function in L 2 (0, 1) with respect to some particular frame in L 2 (0, 1). Since any multiply stable generated shift-invariant space is the image of L 2 (0, 1) by means of a bounded invertible operator, the generalized sampling is obtained from some dual frame expansions in L 2 (0, 1). An example in the setting of the Hermite cubic splines is exhibited.

Regular multivariate sampling and approximation in Lp shift-invariant spaces

“…. , r. Based on previous work of the authors [18,19,21], we can state that the entries of the function g l in (8) belong to L 2 (R d ) whenever h l ∈ L 1 (R d ) ∩ L 2 (R d ), l = 1, 2, . .…”

“…Since {(Υ l f )(Mα)} α∈Z d belongs to 1 (Z d ) and d l, j ∈ A, the series in (19) also converges in the norm of A × · · · × A. Indeed, for N ∈ N,…”

“…. , s (which is equivalent to B G < ∞) means that {g l (x)e −2π iα M x } α∈Z d , l=1,2,...,s is a Bessel sequence for the product Hilbert space L 2 [0, 1) d × · · · × L 2 [0, 1) d (r times) (see [19,Lemma 2]). …”

“…Searching for reconstruction functions S l with compact support as in [19] we obtain in V p ϕ 1 ,ϕ 2 the following sampling formula:…”

Reconstruction of Multiply Generated Splines from Local Average Samples

“…In general, spline spaces yield many advantages in their generation and numerical treatment so that there are many practical applications in signal, image processing, and communication theory. In the literature [1][2][3][4][5][6][7][8] many authors have investigated the generalized sampling technique for multiply generated shift-invariant spaces and spline subspaces. The multiply generated spline space is defined in [5,6] as…”