Invertibility of Laurent operators and shift invariant spaces with finitely many generators

Abstract

In this paper, it is shown that for a fixed $m\in \mathbb{N},$ $\mathbb{Z}/m$ is a stable set of sampling for $V({\phi }^1,{\phi }^2,\dots ,{\phi }^s)$ if and only if there exists $\gamma >0$ such that $|det{\mathrm{\Phi }}^{\mathrm{♯}}\left(x\right)|\ge \gamma$ a.e. on $\mathbb{T}$, where ${\mathrm{\Phi }}^{\mathrm{♯}}$ is a symbol matrix of a block Laurent operator associated with the sample set of ${\phi }^l,l=1,\dots ,s$. A similar result is proved when $r\mathbb{Z},$ where $r=p/q,p,q\in \mathbb{N}$ and $gcd(p,q)=1,$ is a stable set of sampling for $V\left(\phi \right)$ (single generator) as well as $V({\phi }^1,{\phi }^2,\dots ,{\phi }^s)$ (multi generators ). A perturbation of stable set of sampling is also discussed.

COMMUNICATED BY:

• Z. Nashed

Keywords:

• Laurent operator
• nonuniform sampling
• reproducing kernel Hilbert space
• shift invariant space
• Wiener amalgam space

2010 Mathematics Subject Classifications:

• 42C15
• 94A20

Acknowledgments

This work was partially done while the author (S.S.) was visiting Department of Mathematics, IIT Madras, and he gratefully acknowledges the hospitality at Chennai.

Disclosure statement

No potential conflict of interest was reported by the authors.