Recovery of functions by interpolation and sampling
The paper deals with recovering of an arbitrary element \(x\) in a normed space \(X\) from its values \((\phi_n(x))^\infty_{n=1}\) of a sequence of linear functionals \((\phi_n)_{n=1}^\infty\). The operators \(T_n:\mathbb{C}^n\to X\), \(n=1,2,3,\dots\) are said to be recovery operators, if \(|T_n(\phi_1(x),\dots,\phi_n(x))-x|\to 0\) for all \(x\in X\). The authors give some general results on the existence of recovery operators in normed space with particular reference to minimal norm interpolation. The question of whether recovery operators can be linear is studied as well. This is related to an already-studied case, the disk algebra. Also, the question of sampling of band-limited functions is discussed. The author shows how a result of Beurling can be used to construct recovery operators for sampling even in the presence of noise.
scientific article; zbMATH DE number 902503