Reproducing kernels related to the complex sphere
type: Item
instance of: scholarly article
title: Reproducing kernels related to the complex sphere
zbMATH Open document ID: 1004.46024
author: Keiko Fujita
author: Mitsuo Morimoto
published in: Tokyo Journal of Mathematics
publication date: 2002-01-28T00:00:00Z
review text: The authors study four reproducing kernel Hilbert spaces (RKHS) of complex-valued functions and describe their relationships. In \(C^{n+1}\) \(\widetilde{B}[r]\) denotes the closed Lie ball, and \(\widetilde{B}(r)\) the open Lie ball of radius \(r>0\), and \(\widetilde{S}_{\lambda}\) denotes the complex sphere of radius \(\lambda\in C\). Set \(\widetilde{S}_{\lambda}[r]=\widetilde{S}_{\lambda}\cap \widetilde{B}[r]\), and \(\widetilde{S}_{\lambda,r}=\partial \widetilde{S}_{\lambda}[r]\) the boundary of \(\widetilde{S}_{\lambda}[r]\) when \(|\lambda|\leq r\). The four RKHS are defined as follows. \({h}^2_{\lambda}(\widetilde{B}(r))\), the completion of the space of germs of complex harmonic functions on \(\widetilde{B}[r]\) with respect to (w.r.t.) the form \[ (f,g)_{\widetilde{S}_{\lambda,r}}=\int_{\widetilde{S}_{\lambda,r}}f(z)\overline{g(z)} dz; \] \(H^2(\widetilde{S}_{\lambda}(r))\), the Hardy space of holomorphic functions on \(\widetilde{S}_{\lambda}(r)\); \({\mathcal E}^2(\widetilde{S}_0;\lambda,r)\), the image of \({h}^2_{\lambda}(\widetilde{B}(r))\) under the conical Fourier transform; and \({\mathcal E}^2_{\Delta-\lambda^2}(\widetilde{E};r)\), the image of \(H^2(\widetilde{S}_{\lambda}(r))\) under the Fourier transform. This study continues previous works by the authors, for instance [\textit{J.-M. Bony} (ed.), et al., New trends in microlocal analysis. Tokyo: Springer, 39-58 (1997; Zbl 0877.35082)].
reviewed by: Radu Balan
Mathematics Subject Classification ID: 46E22
zbMATH DE Number: 1523375
zbMATH Keywords: reproducing kernel Hilbert space
zbMATH Keywords: complex sphere
zbMATH Keywords: Hardy space of holomorphic functions
zbMATH Keywords: Fourier transform
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1255958813
OpenAlex ID: W2086967333
DOI: 10.3836/TJM/1255958813
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description: scientific article; zbMATH DE number 1523375