On spaces of analytic functions of several variables
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| Názov: | On spaces of analytic functions of several variables |
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| Autori: | Geche, F. I. |
| Informácie o vydavateľovi: | Polychka ``Karpats'kogo Krayu'', Uzhgorod |
| Predmety: | Isomorphic theory (including renorming) of Banach spaces, Locally convex Fréchet spaces and (DF)-spaces, spaces of analytic functions of several variables, abstract analytic space, locally convex spaces, Analytic subsets and submanifolds, holomorphic functions, topological isomorphism, Topology of analytic spaces, entire functions |
| Popis: | This paper deals with topological isomorphism between some locally convex spaces of holomorphic or entire functions of several complex variables and abstract analytic spaces. Let the series \(\sum_{k\in\mathbb{Z}_{+}^{n}}f_{k}z^{k}=\sum_{k\in K_{f}}f_{k}z^{k_1}_1\ldots z_{n}^{k_{n}}\) \((K_{f}=\{k\in\mathbb{Z}_{+}^{n}:f_{k}\neq 0\})\) have a non-empty convergence domain \(D(f)\subset\mathbb{C}^{n}\), i.e. represents the function \(f=(f_{k})\) holomorphic in \(0\in\mathbb{C}^{n}\). We denote by \(\overline A_0\) the set of all functions \(f\) holomorphic in \(0\). Let \(\Phi=\{\Phi_{k}:\mathbb{R}_{+}\to\overline\mathbb{R}_{+}, k\in\mathbb{Z}_{+}^{n}\}\) be a characteristic, and let \(\chi_{\Phi}^{f}:\Sigma_{+}\to \overline\mathbb{R}_{+}\) be directed \(\Phi\)-characteristic, \(\chi_{\Phi}^{f}(a)=\lim_{\pi(k)\to a}\Phi_{k}(|f_{k}|)\), if \(a\in \Sigma_{K_{f}}\), and \(\chi_{\Phi}^{f}(a)=-\infty\), if \(a\in \Sigma_{+}\setminus\Sigma_{K_{f}}\), \(\pi(k)=k/\|k\|, k\in\mathbb{Z}_{+}^{n}\). The hypersurface \(S_{f}\) of conjugate \((\Phi,\Psi)\)-characteristics of the function \(f\) is defined by \[ S_{f}=\Bigl\{r\in D_{\Psi}:\max_{a\in\Sigma_{+}}\chi_{\Phi}^{f}(a)\nu(\langle\beta(a),\eta^{-1}(r)\rangle)=1\Bigr\}=\partial[\eta(H_{\Phi}^{f}\cap\Delta_{\Psi})], \] where \(\eta\) is a homeomorphism of the convex domain \(\Delta_{\Psi}\subset\mathbb{R}^{n}\) on the domain \(D_{\Psi}\subset\mathbb{R}^{n}\); \(\nu:]a,b[\to \widehat\mathbb{R}\) and \(\beta:\Sigma_{+}\to\mathbb{R}^{n}\setminus\{0\}\) are homeomorphic mappings such that \(\langle\pi(\beta(a)),a\rangle>0,\;a\in \Sigma_{+}\), \(\inf\{\langle\beta(a),\xi\rangle:\xi\in\Delta_{\Psi}\}=a\), \(\sup\{\langle\beta(a),\xi\rangle:\xi\in\Delta_{\Psi}\}=b\), \(H_{\Phi}^{f}:=\bigcap_{00\), where \(d\) is the Euclidean metric. The author proves that the vector spaces \(A_{\Phi}[\Omega]\) and \(A_{\Phi}[\Omega)\) are isomorphic respectively to abstract analytic spaces \(A_{\Phi}(\varphi)\) and \(\overline{A}_{\Phi}(\varphi)\), where \(\varphi(a)=1/\widetilde\nu(\varepsilon h(\varepsilon\beta(a))), a\in\Sigma_{+}\), \(\varepsilon=+1\) or \(\varepsilon=-1\) depending on either function \(\nu\) is increasing or decreasing respectively. If \(\Omega=D_{\Psi}\), then \(\varphi(a)\equiv 0\), if \(\Omega=\emptyset\), then \(\varphi(a)\equiv+\infty\). The author also proves that spaces \(A_{\Phi}[\Omega]\) and \(A_{\Phi}[\Omega)\) are inductive and projective limit of a sequence of Banach spaces respectively. Examples of the corresponding spaces and characteristics are presented. |
| Druh dokumentu: | Article |
| Popis súboru: | application/xml |
| Prístupová URL adresa: | https://zbmath.org/1681801 |
| Prístupové číslo: | edsair.c2b0b933574d..354cac858a0b8dc83aa5b565fd118ad9 |
| Databáza: | OpenAIRE |
Nájsť tento článok vo Web of Science | Abstrakt: | This paper deals with topological isomorphism between some locally convex spaces of holomorphic or entire functions of several complex variables and abstract analytic spaces. Let the series \(\sum_{k\in\mathbb{Z}_{+}^{n}}f_{k}z^{k}=\sum_{k\in K_{f}}f_{k}z^{k_1}_1\ldots z_{n}^{k_{n}}\) \((K_{f}=\{k\in\mathbb{Z}_{+}^{n}:f_{k}\neq 0\})\) have a non-empty convergence domain \(D(f)\subset\mathbb{C}^{n}\), i.e. represents the function \(f=(f_{k})\) holomorphic in \(0\in\mathbb{C}^{n}\). We denote by \(\overline A_0\) the set of all functions \(f\) holomorphic in \(0\). Let \(\Phi=\{\Phi_{k}:\mathbb{R}_{+}\to\overline\mathbb{R}_{+}, k\in\mathbb{Z}_{+}^{n}\}\) be a characteristic, and let \(\chi_{\Phi}^{f}:\Sigma_{+}\to \overline\mathbb{R}_{+}\) be directed \(\Phi\)-characteristic, \(\chi_{\Phi}^{f}(a)=\lim_{\pi(k)\to a}\Phi_{k}(|f_{k}|)\), if \(a\in \Sigma_{K_{f}}\), and \(\chi_{\Phi}^{f}(a)=-\infty\), if \(a\in \Sigma_{+}\setminus\Sigma_{K_{f}}\), \(\pi(k)=k/\|k\|, k\in\mathbb{Z}_{+}^{n}\). The hypersurface \(S_{f}\) of conjugate \((\Phi,\Psi)\)-characteristics of the function \(f\) is defined by \[ S_{f}=\Bigl\{r\in D_{\Psi}:\max_{a\in\Sigma_{+}}\chi_{\Phi}^{f}(a)\nu(\langle\beta(a),\eta^{-1}(r)\rangle)=1\Bigr\}=\partial[\eta(H_{\Phi}^{f}\cap\Delta_{\Psi})], \] where \(\eta\) is a homeomorphism of the convex domain \(\Delta_{\Psi}\subset\mathbb{R}^{n}\) on the domain \(D_{\Psi}\subset\mathbb{R}^{n}\); \(\nu:]a,b[\to \widehat\mathbb{R}\) and \(\beta:\Sigma_{+}\to\mathbb{R}^{n}\setminus\{0\}\) are homeomorphic mappings such that \(\langle\pi(\beta(a)),a\rangle>0,\;a\in \Sigma_{+}\), \(\inf\{\langle\beta(a),\xi\rangle:\xi\in\Delta_{\Psi}\}=a\), \(\sup\{\langle\beta(a),\xi\rangle:\xi\in\Delta_{\Psi}\}=b\), \(H_{\Phi}^{f}:=\bigcap_{00\), where \(d\) is the Euclidean metric. The author proves that the vector spaces \(A_{\Phi}[\Omega]\) and \(A_{\Phi}[\Omega)\) are isomorphic respectively to abstract analytic spaces \(A_{\Phi}(\varphi)\) and \(\overline{A}_{\Phi}(\varphi)\), where \(\varphi(a)=1/\widetilde\nu(\varepsilon h(\varepsilon\beta(a))), a\in\Sigma_{+}\), \(\varepsilon=+1\) or \(\varepsilon=-1\) depending on either function \(\nu\) is increasing or decreasing respectively. If \(\Omega=D_{\Psi}\), then \(\varphi(a)\equiv 0\), if \(\Omega=\emptyset\), then \(\varphi(a)\equiv+\infty\). The author also proves that spaces \(A_{\Phi}[\Omega]\) and \(A_{\Phi}[\Omega)\) are inductive and projective limit of a sequence of Banach spaces respectively. Examples of the corresponding spaces and characteristics are presented. |
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