Spaces of analytic functions of prescribed growth near the boundary

V. V. Napalkov

Izvestiya. Mathematics , Tome 30 (1988) no. 2, pp. 263-281

Résumé

Let $D$ be an arbitrary bounded convex domain in the plane $\mathbf C$. For a certain sequence of convex functions $\varphi=\{\varphi_j\}_{j=1}^\infty$, $\varphi_j(z)\geqslant\varphi_{j+1}(z)$, given on $D$ the space $H_\varphi (D)$ is constructed as the projective limit of the normed spaces $$ H_j(D)=\{f(z)\in H(D):\|f\|_j=\sup_D|f(z)|\exp{(-\varphi_j(z))}\infty\},\qquad j=1,2,\dots, $$ where $H(D)$ is the space of analytic functions on $D$. The space $H_\varphi^*(D)$ is described in terms of Laplace transforms. A special role in this description is played by a generalization, proved in the article, of the Paley–Wiener theorem to the case of spaces of infinitely differentiable functions with prescribed growth near the boundary. The result is used in questions involving expansions of functions in Dirichlet series. Figures: 1. Bibliography: 17 titles.