Avetisyan, K. L. Integral representations in general weighted Bergman spaces. (English) Zbl 1089.30037 Complex Variables, Theory Appl. 50, No. 15, 1151-1161 (2005). This article concerns holomorphic functions in the upper half-plane that are \(p\)th-power integrable (where \(1\leq p<\infty\)) with respect to a positive, continuous weight function that depends only on the distance from the boundary. The weight function is allowed to approach \(0\) at the boundary at an arbitrary rate. Using a fractional integration technique, the author constructs an integral representation of Bergman type for such holomorphic functions. When \(p=2\), the author exhibits an isometric isomorphism between \(L^2({\mathbb R})\) and the space of weighted square-integrable holomorphic functions in the upper half-plane. Reviewer: Harold P. Boas (College Station) Cited in 4 Documents MSC: 30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane 30C40 Kernel functions in one complex variable and applications Keywords:reproducing kernel; fractional integration; Paley-Wiener theorem PDFBibTeX XMLCite \textit{K. L. Avetisyan}, Complex Variables, Theory Appl. 50, No. 15, 1151--1161 (2005; Zbl 1089.30037) Full Text: DOI