A generalization of theorems of Eidelheit and Carleman concerning approximation and interpolation (Q1201283)

Let \(U\) be a nonempty subset of \(R\) and let \(C(U)\) be the space of all complex-valued continuous functions defined on \(U\). Let \(X\) be a complex linear space and suppose that for every \(s\in U\) it is given a complex linear functional \(F_ s\) on \(X\). The system \((X,U,F_ s)\) is said to have the asymptotic approximation property (A) if for every \(f,h\in C(U)\), with \(h(s)>0\), \(s\in U\), there exists \(x\in X\) such that (1) \(| f(s)-F_ s(x)|<h(s)\), \(s\in U\). Taking \(h(s)=\varepsilon>0\), \(s\in U\), one obtains the property (A). If furthermore, for every sequence \((q_ n)\) of distinct points in \(U\), such that every compact subset of \(U\) contains only finitely many points \(q_ n\), and every \(f,h\in C(U)\), with \(h(s)>0\), \(s\in U\), there exists \(x\in X\), satisfying (1) and such that \(F_{q_ n}(x)=f(q_ n)\), \(n\in N\), then the system \((X,U,F_ s)\) is said to have the simultaneous approximation and interpolation property \((A,I)\). The main result of the paper is a functional analysis identity theorem concerning relationship between approximation in normed spaces satisfying some additional restrictions by seminorms and corresponding identity properties of bounded linear functionals on these spaces (the main tool used in the proofs is the Hahn-Banach extension theorem). This result enables the author to extend several results of classical analysis and making their proof easier and more transparent. Among these results one can mention Eidelheit's theorem on the existence of solutions of infinite dimensional systems of linear equations \(\sum_{k=1}^ \infty a_{ik} x_ k=c_ i\), \(i\in N\) [\textit{M. Eidelheit}, Stud. Math. 6, 139-148 (1936; Zbl 0015.35603)], Carleman's theorem on the approximation of functions in \(C(R)\) by entire functions [\textit{T. Carleman}, Ark. Mat. Astron. Phys. B 20, 1-5 (1927; JFM 53.0237.02)] and Pólya's theorem on the problem of moments [\textit{G. Pólya}, C. R. Acad. Sci. 207, No. 4, 708-711 (1938; Zbl 0020.04201)]. For instance, taking \(F_ s(x)=\sum_{k=1}^ \infty a_ k K_ k(s)\), \(K_ k\in C(U)\), for \((a_ k)\) belongong to a given sequence space \(X\), one obtains an extension of Eidelheit's result. For \(K_ k(s)=s^{\lambda_ k}\), with given \(\lambda_ k\geq 0\), one obtains necessary and sufficient conditions for asymptotic approximation and interpolation by Dirichlet series, containing as particular case the approximation by Müntz polynomials. Pólya's moment theorem is obtained taking \(K_ k(s)=s^{m_ k-1}\), \(F_ s(g)=\int_ 0^ \infty g(u) u^ s du\) and \(X\) an appropriate space of entire functions on \([0,\infty)\).