A general sampling theory in the functional Hilbert space induced by a Hilbert space valued kernel

García, Antonio G.; Portal, Alberto
July 2008
Journal of Applied Functional Analysis;Jul2008, Vol. 3 Issue 3, p299
Academic Journal
Let ℍ be a separable Hubert space and Ω a fixed subset of ℝ. Consider an ℍ-valued function K : Ω → ℍ and χ ϵ ℍ. Then, the function ƒχ : Ω → ℂ given by ƒχ(t) := (χ, K(t))ℍ is well-defined. Denote by HK the set of functions obtained in this way. Although a variety of sampling results for HK is known in the literature, there exist some simple examples where they do not apply because the implicit interpolation condition that appears does not adjust the former pattern we need. The main aim of this paper is to obtain a more general sampling result including most of these special cases. In this way, the concept of interpolation condition is redefined and we study how to combine two of them in order to obtain a new sampling result. Some examples are obtained in this new framework.


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