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Reproducing kernels related to the complex sphere. (English) Zbl 1004.46024
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 [J.-M. Bony (ed.), et al., New trends in microlocal analysis. Tokyo: Springer, 39-58 (1997; Zbl 0877.35082)].

46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
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