Sampling in the image of Sobolev Space in $$L^2({\mathbb {R}}, e^{u^2}du)$$ under Schrödinger semigroup $$e^{it\Delta }$$

Abstract

It is known that the image of a Sobolev space $$W^{m,2}({\mathbb {R}})$$, $$m\in {\mathbb {N}}$$ in $$L^2({\mathbb {R}}, e^{u^2}du)$$ under the Schrodinger semigroup $$e^{it\Delta }$$ is a Hilbert space of entire functions $$\mathcal {HL}^2({\mathbb {C}}, u_t^m(z)dz)$$. In this article, it is shown that a lattice Z in the complex plane to be a sampling for the space $$\mathcal {HL}^2({\mathbb {C}}, u_t^m(z)dz)$$ if and only if the lower density $$D^{-}(Z)> \frac{1}{8t^2\pi }$$.