Treffer: modified moment problem in complex domains: Modified moment problem in complex domains

Let \(D\) be a simply connected bounded domain in the complex plane \(\mathbb{C}\). The author studies a moment problem of the following kind: Let \(p\geq 2\) functions \(W_1,\ldots,W_p\) holomorphic and one-to-one in \(D\) and \(p\) functions \(A_0,\ldots,A_{p-1}\) holomorphic (and zero-free) in \(D\) be given. Suppose further that \((a_{nj})^\infty_{n=0}\) are \(p\) sequences of complex numbers and that \(\ell_0,\ldots,\ell_{p-1}\) are nonnegative integers. Does there exist a function \(g\) holomorphic in a (simply connected) neighbourhood \(U\) of \(\mathbb{C}_\infty\backslash D\), where \(\mathbb{C}_\infty\) is the extended plane, vanishing at \(\infty\), and such that \[ \frac{1}{2\pi i} \int\limits_\Gamma W^{np+\ell_j}_j (z) A_j (z) g(z) dz \, = \, a_{nj} \] for \(n = 0,1,2,\ldots\) and \(j = 0,1,\ldots,p-1\), where \(\Gamma\) is an arbitrary closed rectifiable Jordan curve in \(U\cap D\)? Under a certain regularity condition on the sequences \((a_{nj})\) he gives a characterization of the solvability. Moreover, the author applies his main result to the Abel-Goncharov problem for entire functions of exponential type. According to Köthe-duality, the non-trivial solvability of the above moment problem for \(a_{nj} \equiv 0\) is equivalent to the non-completeness of the functions \(W^{np+\ell_j}_j A_j \; (j = 0,\ldots,p-1; n = 0,1,\ldots)\) in \(A(D)\), the space of functions holomorphic in \(D\) with the usual topology. This fact is used to prove a completeness result for \(A(D_R)\), where \(D_R\) is the disk of radius \(R\) around \(0\).