A generalization of theorems of Eidelheit and Carleman concerning approximation and interpolation

Abstract

We prove necessary and sufficient conditions for linear operators to approximate and interpolate unbounded continuous functions on certain subsets U ⊆ (−∞, ∞). The main application of our general theory is to simultaneous asymptotic approximation and interpolation by function series. Special cases of our results are a sharpened version of a theorem of Eidelheit for the solubility of infinite systems of linear equations and a generalization of a theorem of Carleman concerning the asymptotic approximation and interpolation of continuous functions by entire functions on the real axis. Moreover we can apply our general theorems to a moment problem of Pólya and to asymptotic approximation and interpolation by Dirichlet series. Our general approach to such problems is based on the use of certain complete approximation systems and on an essential identity theorem of functional analysis concerning approximations in normed linear spaces with certain additional restrictions by seminorms.