Abstract
We consider translates of functions in L 2 ( R d ) along an irregular set of points, that is, { ϕ ( · − λ k ) } k ∈ Z —where ϕ is a bandlimited function. Introducing a notion of pseudo-Gramian function for the irregular case, we obtain conditions for a family of irregular translates to be a Bessel, frame or Riesz sequence. We show the connection of the frame-related operators of the translates to the operators of exponentials. This is used, in particular, to find for the first time in the irregular case a representation of the canonical dual as well as of the equivalent Parseval frame—in terms of its Fourier transform.
Highlights
Frames of translates, i.e., frame sequences originated by the shifts of a given, fixed function, are an important mathematical background for Gabor [1,2,3], wavelet [4,5] and sampling theory [6,7].One generating function φ is shifted to create the analyzing family of elements, {φ (· − λk )}k∈Z for a sequence Λ = {λk }k∈Z
For the case of irregular frames of translates, where we have no group structure, we introduce the following notion
We arrive to the relation between the canonical duals of frame sequences of exponentials and frame sequences of translates stated in Theorem 2
Summary
I.e., frame sequences originated by the shifts of a given, fixed function, are an important mathematical background for Gabor [1,2,3], wavelet [4,5] and sampling theory [6,7]. One generating function φ is shifted to create the analyzing family of elements, {φ (· − λk )}k∈Z for a sequence Λ = {λk }k∈Z These systems play a main role in the theory of shift invariant spaces (SIS) [8,9,10,11], which is very useful for the modeling of problems in signal processing, and is central in approximation. This permits us to study Bessel, frame and Riesz properties of irregular frames of translates, and give formulas for the canonical dual, and for equivalent Parseval frames
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