Study of a Hilbert space of holomorphic functions (Q1977165)

In the paper [Commun. Pure Appl. Math. 14, 187-214 (1961; Zbl 0107.09102)], \textit{V. Bargmann} considered the Hilbert space \({\mathcal F}_1\) consisting of all entire functions \(f(z),\;z\in {\mathbb C}\), such that \[ \|f\|^2=\int_{\mathbb C}|f(z)|^2 d\nu <\infty, \tag{*} \] where \(d\nu =\pi^{-1}\exp (-|z|^2) dx dy\;(z=x+iy)\) is the Gaussian measure on the plane \({\mathbb C}\). Among other things, he showed that \(f\in {\mathcal F}_1\) if and only if \[ \sum_{k=0}^\infty k! |a_k|^2<\infty, \tag{**} \] where \(f(z)=\sum_{k=0}^\infty a_kz^k\) is the Taylor expansion of \(f\), and that the sequence of entire functions \(\{ z^k/\sqrt{k!}:k=0,1,2,\ldots\}\) forms an orthonormal basis for \({\mathcal F}_1\). In the paper under review, the author develops an analogous theory for the Hilbert space \({\mathcal U}(\Omega)\) of holomorphic functions on the domain \(\Omega :=\{ z\in {\mathbb C}:|\arg z|<\pi\}\), for which condition (*) is satisfied. This space contains the Bargmann space \({\mathcal F}_1\), and also many important functions like the principle branches of \(\log z\) and arbitrary powers \(z^\alpha\) of \(z\in {\mathbb C}\). In particular, he constructs an orthonormal basis for \({\mathcal U}(\Omega)\), and defines a family of Lebesgue spaces \({\mathcal L}_\alpha\;(\alpha \geq 0)\), consisting of functions on \((0,\infty)\) which are square integrable with respect to a certain weighted measure \(d\mu_\alpha\), and playing the same role as the space of coefficient sequences \((a_0,a_1,a_2,\ldots)\) satisfying condition (**) for the case of the Bargmann space \({\mathcal F}_1\), together with the corresponding family of linear transformations \(M_\alpha :{\mathcal L}_\alpha\to {\mathcal U}(\Omega)\;(\alpha\geq 0)\), playing the same role as the transformation \((a_0,a_1,a_2,\ldots)\mapsto\sum_{k=0}^\infty a_kz^k\) in Bargmann's case. The author studies properties of the \(M_\alpha\)'s, and obtains a certain approximation result in the \(L^2\)-norm for functions in \({\mathcal U}(\Omega)\) by means of integrals of the form \[ \int_0^\infty \varphi (t)e^{-t}t^\alpha z^t dt\quad (z\in\Omega), \] where \(\varphi\in {\mathcal L}_\alpha\) is a real analytic function on \((0,\infty)\).