Narrow Results

A subspace V of L₂(ℝ) is called shift-invariant if it is the closed linear span of integer-shifted copies of a single function.

As a complement to classical analysis techniques for sampling in such spaces, we propose a method which is based on a simple interpolation estimate of a certain coefficient mapping. Then we use this method to derive both new results and relatively simple proofs of some previously known results. Among these are some results of rather general nature and some more specialized results for B-spline wavelets.

The main problem under study is to find a shift x₀ and an upper bound δ such that any function f ∈ V can be reconstructed from a sequence of sample values (f(x₀ + k + δ_k))_k∈ℤ, either when all δ_k = 0 or in the irregular sampling case with an upper bound sup_k|δ_k| < δ.

We consider irregular sampling in shift invariant spaces V of higher dimensions. The problem that we address is: find ε so that given perturbations (λ_k) satisfying sup|λ_k| < ε, we can reconstruct an arbitrary function f of V as a Riesz basis expansions from its irregular sample values f(k + λ_k). A framework for dealing with this problem is outlined and in which one can explicitly calculate sufficient limits ε for the reconstruction. We show how it works in two concrete situations.

In this paper we study various perturbation techniques in the context of irregular spline-type spaces. We first present the sampling problem in this general setting and prove a general result on the possibility of perturbing sampling sets. This result can be regarded as a spline-type space analogue in the spirit of Kadec's Theorem for bandlimited functions (see Refs. 14 and 15). We further derive some quantitative estimates on the amount by which a sampling set can be perturbed, and finally prove a result on the existence of optimal perturbations (with the stability of reconstruction being the optimality criterion). Finally, the techniques developed in the earlier parts of the paper are used to study the problem of disturbing a basis for a spline-type space, in order to derive a sufficient criterion for a space generated by irregular translations to be a spline-type space.

In the first part of this paper we study the problem of recovering bandlimited signals by irregular sampling, that is from their values at irregularly distributed points. We obtain a sufficient condition for a sampling sequence to fit any bandlimited signal with an arbitrary bandwidth. The sampling functions are made of eigensolutions of a singular Sturm-Liouville operator whose spectrum is the set of sample points. The main tool is the Gelfand-Levitan theory which is used to produce such an operator. The second part deals with sampling in Hardy spaces. There we prove a new sampling formula and show that Shannon type sampling formulae cannot hold for functions in the Hardy space . As an example, we work out a new series representation of the Riemann zeta function in the half-plane ℜ(s) > ½. Upper bounds for the truncation and amplitude errors for the sampling formulae are also provided.