Recent zbMATH articles in MSC 30H20https://www.zbmath.org/atom/cc/30H202022-05-16T20:40:13.078697ZWerkzeugBounded extremal problems in Bergman and Bergman-Vekua spaceshttps://www.zbmath.org/1483.300892022-05-16T20:40:13.078697Z"Delgado, Briceyda B."https://www.zbmath.org/authors/?q=ai:delgado.briceyda-b"Leblond, Juliette"https://www.zbmath.org/authors/?q=ai:leblond.julietteSummary: We analyze Bergman spaces \(A_f^p(\mathbb{D})\) of generalized analytic functions of solutions to the Vekua equation \(\bar{\partial}w = (\bar{\partial}f/f)\bar{w}\) in the unit disc of the complex plane, for Lipschitz-smooth non-vanishing real valued functions \(f\) and \(1<p<\infty\). We consider a family of bounded extremal problems (best constrained approximation) in the Bergman space \(A^p(\mathbb{D})\) and in its generalized version \(A^p_f(\mathbb{D})\), that consists in approximating a function in subsets of \(\mathbb{D}\) by the restriction of a function belonging to \(A^p(\mathbb{D})\) or \(A^p_f(\mathbb{D})\) subject to a norm constraint. Preliminary constructive results are provided for \(p = 2\).Composition of analytic paraproductshttps://www.zbmath.org/1483.300962022-05-16T20:40:13.078697Z"Aleman, Alexandru"https://www.zbmath.org/authors/?q=ai:aleman.alexandru"Cascante, Carme"https://www.zbmath.org/authors/?q=ai:cascante.carme"Fàbrega, Joan"https://www.zbmath.org/authors/?q=ai:fabrega.joan"Pascuas, Daniel"https://www.zbmath.org/authors/?q=ai:pascuas.daniel"Peláez, José Ángel"https://www.zbmath.org/authors/?q=ai:pelaez.jose-angelSummary: For a fixed analytic function \(g\) on the unit disc \(\mathbb{D}\), we consider the analytic paraproducts induced by \(g\), which are defined by \(T_g f(z)=\int_0^z f(\zeta)g^\prime(\zeta)d\zeta\), \(S_g f(z)=\int_0^z f^\prime(\zeta) g(\zeta)d\zeta\), and \(M_g f(z)=f(z) g(z)\). The boundedness of these operators on various spaces of analytic functions on \(\mathbb{D}\) is well understood. The original motivation for this work is to understand the boundedness of compositions of two of these operators, for example \(T_g^2\), \(T_gS_g\), \(M_g T_g\), etc. Our methods yield a characterization of the boundedness of a large class of operators contained in the algebra generated by these analytic paraproducts acting on the classical weighted Bergman and Hardy spaces in terms of the symbol \(g\). In some cases it turns out that this property is not affected by cancellation, while in others it requires stronger and more subtle restrictions on the oscillation of the symbol \(g\) than the case of a single paraproduct.Norms of inclusions between some spaces of analytic functionshttps://www.zbmath.org/1483.300982022-05-16T20:40:13.078697Z"Llinares, Adrián"https://www.zbmath.org/authors/?q=ai:llinares.adrianSummary: The inclusions between the Besov spaces \(B^q\), the Bloch space \(\mathcal{B}\) and the standard weighted Bergman spaces \(A^p_{\alpha}\) are completely understood, but the norms of the corresponding inclusion operators are in general unknown. In this work, we compute or estimate asymptotically the norms of these inclusions.Complete interpolating sequences for small Fock spaceshttps://www.zbmath.org/1483.300992022-05-16T20:40:13.078697Z"Omari, Youssef"https://www.zbmath.org/authors/?q=ai:omari.youssefSummary: We give a characterization of complete interpolating sequences for the Fock spaces \(\mathcal{F}_\varphi^p\), \(1\leq p< \infty\), where \(\varphi(z)=\alpha(\log^+|z|)^2\), \(\alpha > 0\). Our results are analogous to the classical Kadets-Ingham's 1/4-Theorem on perturbation of Riesz bases of complex exponentials, and they answer a question asked by \textit{A. Baranov} et al. [J. Math. Pures Appl. (9) 103, No. 6, 1358--1389 (2015; Zbl 1315.30009)].On the Fock kernel for the generalized Fock space and generalized hypergeometric serieshttps://www.zbmath.org/1483.460242022-05-16T20:40:13.078697Z"Park, Jong-Do"https://www.zbmath.org/authors/?q=ai:park.jong-doSummary: In this paper, we compute the reproducing kernel \(B_{m, \alpha}(z, w)\) for the generalized Fock space \(F_{m, \alpha}^2(\mathbb{C})\). The usual Fock space is the case when \(m=2\). We express the reproducing kernel in terms of a suitable hypergeometric series \({}_1 F_q\). In particular, we show that there is a close connection between \(B_{4, \alpha}(z, w)\) and the error function. We also obtain the closed forms of \(B_{m, \alpha}(z, w)\) when \(m=1,2/3,1/2\). Finally, we also prove that \(B_{m, \alpha}(z, z)\sim e^{\alpha |z|^m} |z|^{m-2}\) as \(|z|\longrightarrow\infty\).Some essentially normal weighted composition operators on the weighted Bergman spaceshttps://www.zbmath.org/1483.470472022-05-16T20:40:13.078697Z"Fatehi, Mahsa"https://www.zbmath.org/authors/?q=ai:fatehi.mahsa"Shaabani, Mahmood Haji"https://www.zbmath.org/authors/?q=ai:shaabani.mahmood-hajiSummary: First of all, we obtain a necessary and sufficient condition for a certain operator \(T_{w}C_{\varphi}\) to be compact on \(A^{2}_{\alpha}\). Next, we give a short proof for Proposition 2.5 which was proved by \textit{B. D. MacCluer} et al. [Complex Var. Elliptic Equ. 58, No. 1, 35--54 (2013; Zbl 1285.47031)]. Then, we characterize the essentially normal weighted composition operators \(C_{\psi, \varphi}\) on the weighted Bergman spaces \(A^{2}_{\alpha}\), when \(\varphi \in \mathrm{LFT} (\mathbb D)\) is not an automorphism and \(\psi \in H^\infty\) is continuous at a point \(\zeta\) which \(\varphi\) has a finite angular derivative. After that, we find some non-trivially essentially normal weighted composition operators, when \(\varphi \in \mathrm{LFT} (\mathbb D)\) is not an automorphism. In the last section, for \(\varphi \in \mathrm{AUT} (\mathbb D)\) and \(\psi \in {A} (\mathbb D)\), we characterize the essentially normal weighted composition operators \(C_{\psi, \varphi}\) on \(A^{2}_{\alpha}\) and investigate some essentially normal weighted composition operators \(C_{\psi, \varphi}\) on \(H^2\) and \(A^{2}_{\alpha}\). Finally, we find some non-trivially essentially normal weighted composition operators \(C_{\psi, \varphi}\) on \(H^2\) and \(A^{2}_{\alpha}\), when \(\varphi \in \mathrm{AUT} (\mathbb D)\) and \(\psi \in {A} (\mathbb D)\).Co-spherical electronic configuration of the helium-like atomic systemshttps://www.zbmath.org/1483.811622022-05-16T20:40:13.078697Z"Liverts, Evgeny Z."https://www.zbmath.org/authors/?q=ai:liverts.evgeny-zSummary: The properties of a special configuration of a helium-like atomic system, when both electrons are on the surface of a sphere of radius \(r\), and angle \(\theta\) characterizes their positions on sphere, are investigated. Unlike the previous studies, \(r\) is considered as a quantum mechanical variable but not a parameter. It is important that the ``co-spherical'' and the ``collinear'' configuration are coincident in two points. For \(\theta=0\) one obtains the state of the electron-electron coalescence, whereas the angle \(\theta=\pi\) characterizes the \textbf{e-n-e} configuration when the electrons are located at the ends of the diameter of sphere with the nucleus at its center. The Pekeris-like method representing a fully three-body variational technique is used for the expedient calculations. Some interesting features of the expectation values representing the basic characteristics of the ``co-spherical'' electronic configuration are studied. The unusual properties of the expectation values of the operators associated with the kinetic and potential energy of the two-electron atom/ion possessing the ``co-spherical'' configuration are found. Refined formulas for calculations of the two-electron Fock expansion by the Green's function approach are presented. The model wave functions of high accuracy describing the ``co-spherical'' electronic configuration are obtained. All results are illustrated in tables and figures.