Hilbert spaces of holomorphic Dirichlet series and applications to convolution equations
Convolution equations in spaces of holomorphic functions on convex domains \(\Omega\subset \mathbb{C}^N\) have been studied for a long time, starting with Ehrenpreis and Malgrange (for \(\Omega= \mathbb{C}^N\)). Here the author is interested in the case that \(\Omega\) is bounded and convex and that the convolution operator \(M_\mu\) is induced by a nonzero analytic functional \(\mu\) which is carried by a convex compact set \(K\subset \mathbb{C}^N\). By the theory of representing systems [see \textit{Yu. F. Korobejnik}, Russian Math. Surveys 36, No. 1, 75-137 (1981; Zbl 0483.30003)] every holomorphic function \(f\) on \(\Omega\) can be represented as a Dirichlet series \(f(z)= \sum_{k=1}^\infty c_ke^{\langle\lambda^k,z\rangle}\) with \(\Lambda= (\lambda^k)_k\subset \mathbb{C}^N\), but this representation is not unique. Therefore, for solvable convolution equations \(M_\mu[f]=g\), both the right-hand side \(g\) and the solution \(f\) can be represented in the form of Dirichlet series in different ways, and the question arises which is the `best' representation. The aim of the present paper is to solve this question in the context of Hilbert spaces \(X_\beta (\Lambda,\Omega)\) of holomorphic Dirichlet series \(f\) as above, where \(\lim_{k\to\infty} \log k/|\lambda^k|=0\), \(\beta= (\beta_k)_k\) is a sequence of real numbers, and \(f\) belongs to \(X_\beta (\Lambda, \Omega)\) if \(\sum_{k=1}^\infty|c_k|^2 e^{2\beta_k} <+\infty\). The author does not directly work with the function spaces \(X_\beta (\Lambda, \Omega)\), but with the corresponding sequence spaces \(X_\beta\) of the coefficients \(c_k\) of the Dirichlet series \(f\). The point here is to choose the sequence of coefficients in the representation by Dirichlet series (in a uniquely determined way) such that it has minimum Hilbert norm.
scientific article; zbMATH DE number 1005569
scientific article