Complete interpolating sequences for the Gaussian shift-invariant space

Abstract

We give a full description of complete interpolating sequences for the shiftinvariant space generated by the Gaussian. As a consequence, we rederive the known density conditions for sampling and interpolation. Main resultsConsider the shift-invariant space of functions on R with Gaussian generator g(z) = e −az 2 for a > 0 defined asWe consider the space V 2 as a subspace of L 2 (R) with the usual L 2 -norm.The space V 2 belongs to the general family of shift-invariant spaces. Given a generator g ∈ L 2 (R), such a space is defined asThe primary example is the classical Paley-Wiener space P W = {f ∈ L 2 (R) : supp f ⊆ [−1/2, 1/2]}, which, by the sampling theorem of Shannon-Whittaker-Kotelnikov, can be identified with the shift-invariant space V 2 ( sin πx πx ). In signal processing [1] shift-invariant spaces are often taken as a substitute for the Paley-Wiener space. A unifying feature of both V 2 (Gaussian generator) and P W (sinc-generator) is the fact that both spaces can be viewed as spaces of entire functions by interpreting the variable z to be in C.It is easy to see that the norm equivalence f L 2 (R) ≍ (c n ) ℓ 2 (Z) holds on V 2 with some absolute constants. In what follows it will be convenient for us to work with the second quantity and so we put f V 2 := (c n ) ℓ 2 (Z) , and we often identify V 2 with ℓ 2 (Z) via the mapping (c n ) → f (z) = n∈Z c n e −(z−n) 2 .A sequence Λ = {λ n } n∈Z ⊂ R is said to be sampling for V 2 , if