Sampling in shift-invariant spaces of functions with polynomial growth

Abstract

Under suitable conditions, a function f(t) in a principal shift-invariant space can be recovered from its uniform samples { f(n)} _n∈ℤ with a simple sampling formula. Provided that the generator ϕ has compact support, we consider the sampling problem in the bigger space V _ϕ ≔ {∑a _n ϕ(t − n) : a _n  ∈ ℂ^ℤ}. In this space, there exist infinite functions with the same samples y _n  = f(n). We show that polynomial growth conditions give the uniqueness: if y _n has polynomial growth, there is a unique function of polynomial growth f ∈ V _ϕ, satisfying f(n) = y _n , n ∈ ℤ. This function is given by the known sampling formula. The same result is proved also when we consider average samples y _n  = f ∗ h(n).

Keywords:

• interpolation
• shift-invariant spaces
• average sampling
• cardinal splines

AMS Subject Classifications::

• 41A15
• 65D07
• 94A20

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Erratum

Acknowledgements

This work has been supported by the grant MTM2009–08345 from the D.G.I. of the Spanish Ministerio de Ciencia y Tecnología.