Recent zbMATH articles in MSC 32Ahttps://zbmath.org/atom/cc/32A2024-05-13T19:39:47.825584ZWerkzeugOn formal solutions of the Hörmander's initial-boundary value problem in the class of Laurent serieshttps://zbmath.org/1532.320022024-05-13T19:39:47.825584Z"Leinartas, Evgeny K."https://zbmath.org/authors/?q=ai:leinartas.evgenii-konstantinovich"Yakovleva, Tatiana I."https://zbmath.org/authors/?q=ai:yakovleva.tatiana-iSummary: We define a derivation of the ring of Laurent series with supports in rational cones and prove existence and uniqueness of a solution to an analog of one initial-boundary value problem of Hörmander for polynomial differential operators with constant coefficients in the class of formal Laurent series.Difference and primitive operators on the Dunkl-type Fock SPACE \(\mathscr{F}_{\alpha }(\mathbb{C}^d)\)https://zbmath.org/1532.320032024-05-13T19:39:47.825584Z"Soltani, Fethi"https://zbmath.org/authors/?q=ai:soltani.fethi"Nenni, Meriem"https://zbmath.org/authors/?q=ai:nenni.meriemSummary: In 1961, \textit{V. Bargmann} [Commun. Pure Appl. Math. 14, 187--214 (1961; Zbl 0107.09102)] introduced the classical Fock space \(\mathscr{F}(\mathbb{C}^d)\) and in 1984, Cholewinski [SIAM J. Math. Anal. 15, 177--202 (1984; Zbl 0596.46017)] introduced the generalized Fock space \(\mathscr{F}_{\alpha ,e}(\mathbb{C}^d)\). These two spaces are the aim of many works, and have many applications in mathematics, in physics, and in quantum mechanics. In this work, we introduce and study the Fock space \(\mathscr{F}_{\alpha }(\mathbb{C}^d)\) associated to the Dunkl operators \(T_{\alpha_j}\) with \(\alpha_j>-1/2\) for all \(j=1,\ldots ,d\). This space is an extension of the Dunkl-type Fock space \(\mathscr{F}_{\alpha }(\mathbb{C})\) constructed by the first author and \textit{F. Soltani} [J. Math. Anal. Appl. 270, No. 1, 92--106 (2002; Zbl 1012.46033)]. We prove that the space \(\mathscr{F}_{\alpha }(\mathbb{C}^d)\) is a Hilbert space with reproducing kernel. Next, we give an application of the classical theory of reproducing kernels to the Tikhonov regularization problem for the bounded linear operator \(\mathscr{L}:\mathscr{F}_{\alpha }(\mathbb{C}^d)\rightarrow \mathscr{H} \), where \(\mathscr{H}\) is a Hilbert space. Finally, we come up with some results regarding the extremal functions, when \(\mathscr{L}\) is the difference operator and the primitive operator, respectively.Regularity of the \(p\)-Bergman kernelhttps://zbmath.org/1532.320042024-05-13T19:39:47.825584Z"Chen, Bo-Yong"https://zbmath.org/authors/?q=ai:chen.bo-yong"Xiong, Yuanpu"https://zbmath.org/authors/?q=ai:xiong.yuanpuSummary: We show that the \(p\)-Bergman kernel \(K_p(z)\) on a bounded domain \(\Omega\) is of locally \(C^{1, 1}\) for \(p \geq 1\).The proof is based on the locally Lipschitz continuity of the off-diagonal \(p\)-Bergman kernel \(K_p(\zeta, z)\) for fixed \(\zeta\in\Omega\). Global irregularity of \(K_p(\zeta, z)\) is presented for some smooth strongly pseudoconvex domains when \(p \gg 1\). As an application of the local \(C^{1, 1}\)-regularity, an upper estimate for the Levi form of \(\log K_p(z)\) for \(1 < p < 2\) is provided. Under the condition that the hyperconvexity index of \(\Omega\) is positive, we obtain the log-Lipschitz continuity of \(p\mapsto K_p(z)\) for \(1 \leq p \leq 2\).\(L^p\)-boundedness \((1 \leq p \leq \infty)\) for Bergman projection on a class of convex domains of infinite type in \(\mathbb{C}^2\)https://zbmath.org/1532.320052024-05-13T19:39:47.825584Z"Ha, Ly Kim"https://zbmath.org/authors/?q=ai:ha.ly-kimSummary: The main purpose of this paper is to show that over a large class of bounded domains \(\Omega \subset \mathbb{C}^2\), for \(1 < p < \infty\), the Bergman projection \(\mathcal{P}\) is bounded from \(L^p (\Omega, dV)\) to the Bergman space \(A^p (\Omega)\); from \(L^\infty (\Omega)\) to the holomorphic Bloch space \(\mathrm{BlHol}(\Omega)\); and from \(L^1 (\Omega, P(z, z) dV)\) to the holomorphic Besov space \(\text{Besov}(\Omega)\), where \(P(\zeta, z)\) is the Bergman kernel for \(\Omega\).Construction of Szegő and Poisson kernels in convex domainshttps://zbmath.org/1532.320062024-05-13T19:39:47.825584Z"Myslivets, Simona G."https://zbmath.org/authors/?q=ai:myslivets.simona-glebovnaSummary: In this paper, we construct Szegő and Poisson kernels in convex domains in \(\mathbb{C}^n\) and study their properties.Decomposition of functions of finite analytical complexityhttps://zbmath.org/1532.320072024-05-13T19:39:47.825584Z"Beloshapka, Valery K."https://zbmath.org/authors/?q=ai:beloshapka.valerii-kSummary: Functions of two variables can be obtained from functions of one variable by substitutions and additions. Each composition scheme corresponds to a class of functions of two variables, such that they can be represented as a composition with this scheme. It was shown that each class consists of analytical solutions of a certain system of differential polynomials (equations of the class). The paper describes an algorithm for constructing a system of equations of the scheme and for obtaining function representation in the form of a composition with this scheme.Second main theorems and algebraic dependence of meromorphic mappings on parabolic manifolds with moving targetshttps://zbmath.org/1532.320112024-05-13T19:39:47.825584Z"Si Duc Quang"https://zbmath.org/authors/?q=ai:si-duc-quang."Nguyen Van An"https://zbmath.org/authors/?q=ai:nguyen-van-an."Pham Duc Thoan"https://zbmath.org/authors/?q=ai:pham-duc-thoan.Summary: The purpose of this paper is twofold. The first is to give some forms of the second main theorem in Nevanlinna theory for meromorphic mappings from parabolic manifolds intersecting moving targets in general position with truncated counting functions, which are improvements of some recent results. The second is to apply the above forms to the proof of an algebraic dependence theorem for meromorphic mappings on parabolic manifolds sharing moving targets regardless of multiplicity.Three-jets determinations of normalized proper holomorphic maps from \(\mathbb{H}_n\) into \(\mathbb{H}_{3n-2}\)https://zbmath.org/1532.320122024-05-13T19:39:47.825584Z"Gul, Nuray"https://zbmath.org/authors/?q=ai:gul.nuray"Ji, Shanyu"https://zbmath.org/authors/?q=ai:ji.shanyu"Yin, Wanke"https://zbmath.org/authors/?q=ai:yin.wankeSummary: We derive the explicit formula for the normalized rational proper holomorphic maps of \(Rat(\mathbb{H}_n, \mathbb{H}_{3n-2})\). As a consequence, we prove that these maps are determined by their three jets.Weighted Bergman kernels for nearly holomorphic functions on bounded symmetric domainshttps://zbmath.org/1532.320172024-05-13T19:39:47.825584Z"Engliš, Miroslav"https://zbmath.org/authors/?q=ai:englis.miroslav"Youssfi, El-Hassan"https://zbmath.org/authors/?q=ai:youssfi.el-hassan"Zhang, Genkai"https://zbmath.org/authors/?q=ai:zhang.genkaiSummary: We identify the standard weighted Bergman kernels of spaces of nearly holomorphic functions, in the sense of Shimura, on bounded symmetric domains. This also yields a description of the analogous kernels for spaces of ``invariantly-polyanalytic'' functions -- a generalization of the ordinary polyanalytic functions on the ball which seems to be the most appropriate one from the point of view of holomorphic invariance. In both cases, the kernels turn out to be given by certain spherical functions, or equivalently Heckman-Opdam hypergeometric functions, and a conjecture relating some of these to a Faraut-Koranyi hypergeometric function is formulated based on the study of low rank situations. Finally, analogous results are established also for compact Hermitian symmetric spaces, where explicit formulas in terms of multivariable Jacobi polynomials are given.Mean convergence of Fourier-Akhiezer-Chebyshev serieshttps://zbmath.org/1532.420322024-05-13T19:39:47.825584Z"Bello-Hernández, Manuel"https://zbmath.org/authors/?q=ai:bello-hernandez.manuel"del Campo López, Alejandro"https://zbmath.org/authors/?q=ai:del-campo-lopez.alejandroFor \(a\in (0, \pi)\) let consider the interval \((a, 2\pi-a)\) and the weight function \(\omega_{a} (\theta) = \frac{\sin(a/2)}{2 \sin(\theta/2) \sqrt{\cos^{2}(a/2) - \cos^{2}(\theta/2)}} \textbf{1}_{(a, 2\pi-a)}.\) Here \(\textbf{1}_{A}\) denotes the characteristic function of the set \(A.\) This weight is known in the literature as the Akhiezer-Chebyshev weight, supported on the arc of the unit circle \(\Delta_{a}= \{ e^{i\theta}, a < \theta < 2\pi- a\}.\) It was introduced in [\textit{N. I. Akhiezer}, Sov. Math., Dokl. 1, 31--34 (1960; Zbl 0129.28305); translation from Dokl. Akad. Nauk SSSR 130, 247--250 (1960)]
Let \(\{\varphi_{n}(z)\}_{n\geq 0}\) be the sequence of orthonormal polynomials with respect to the inner product \(\langle f, g \rangle= \int _{a}^{2\pi-a}f(e^{i\theta}) \overline{g( e^{i\theta})}\omega_{a}(\theta) d\theta.\) These polynomials have been studied in [\textit{L. Golinskii}, J. Approx. Theory 95, No. 2, 229--263 (1998; Zbl 0916.42016)].
Let \( L^{p}((a, 2\pi-a),\omega_{a}), 1 <p <\infty, \) be the linear space of functions \(f: (a, 2\pi-a) \rightarrow \mathbb{C}\) such that \(\int _{a}^{2\pi-a}| f( e^{i\theta})|^{p} \omega_{a} (\theta) d\theta < \infty.\) In the paper under review the authors prove that for a function \(f\) in such a space and if \(S_{n}( f; z) =\sum_{k=0}^{n} \langle f, \varphi_{k} \rangle \varphi_{k}(z)\) denotes its \(n\)-th partial Fourier sum in terms of the sequence of orthonormal polynomials \(\{\varphi_{n}(z)\}_{n\geq0},\) then you get the mean convergence of Fourier-Akhiezer-Chebyshev series , i. e., \(\lim_{n\rightarrow \infty} \int _{a}^{2\pi-a} | f(e^{i\theta})- S_{n}( f; e^{i\theta})|^ {p} \omega_{a}(\theta) d\theta=0.\) The Banach-Steinhaus theorem is used to prove the existence of a constant \(C,\) independent of \(f,\) such that for every \(n\) the inequality \(\int _{a}^{2\pi-a} |S_{n}( f; e^{i\theta})|^ {p} \omega_{a}(\theta) d\theta \leq C \int _{a}^{2\pi-a} | f(e^{i\theta})|^{p} \omega_{a}(\theta) d\theta\) holds. Then, as a consequence of the Szegő-Kolmogorov-Krein's theorem, the algebraic polynomials are dense in \( L^{p}((a, 2\pi-a),\omega_{a})\) and the statement follows.
For the above weight function, an estimate for the corresponding sequence of para-orthonormal polynomials as well as a weighted inequality for the Hilbert transform on an arc of the unit circle are deduced as auxiliary results. Finally, the mean convergence of the Fourier series in terms of orthonormal polynomials with respect to a perturbation of the Akhiezer-Chebyshev weight times a Lipschitz function is obtained.
Reviewer: Francisco Marcellán (Leganes)Continuous dependence of Szegő kernel on a weight of integrationhttps://zbmath.org/1532.460262024-05-13T19:39:47.825584Z"Żynda, Tomasz Łukasz"https://zbmath.org/authors/?q=ai:zynda.tomasz-lukasz"Pasternak-Winiarski, Zbigniew"https://zbmath.org/authors/?q=ai:pasternak-winiarski.zbigniew"Sadowski, Jacek Józef"https://zbmath.org/authors/?q=ai:sadowski.jacek-jozef"Krantz, Steven George"https://zbmath.org/authors/?q=ai:krantz.steven-georgeSummary: The weighted Szegő kernel was investigated in a few papers (see
[\textit{Z.~Nehari}, J. Anal. Math. 2, 126--149 (1952; Zbl 0049.17603)],
[\textit{Yu.~E. Alenitsyn}, Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 24, 16--28 (1972; Zbl 0263.30010)],
[\textit{M.~Uehara} and \textit{S.~Saitoh}, Math. Japon. 29, 887--891 (1984; Zbl 0561.30007)],
[\textit{M.~Uehara}, Math. Japon. 42, No. 3, 459--469 (1995; Zbl 0839.30005)]).
In all of these, however, only continuous weights were considered. The aim of this paper is to show that the Szegő kernel depends in a continuous way on a weight of integration in the case when the weights are not necessarily continuous. A topology on the set of admissible weights will be constructed and Pasternak's theorem (see
[\textit{Z.~Pasternak-Winiarski}, Stud. Math. 128, No.~1, 1--17 (1998; Zbl 0910.46013)])
on the dependence of the orthogonal projector on a deformation of an inner product will be used in the proof of the main theorem.Inaccessibility regions for the sterile neutrino searches at the long baseline experimentshttps://zbmath.org/1532.810802024-05-13T19:39:47.825584Z"Kaur, Daljeet"https://zbmath.org/authors/?q=ai:kaur.daljeet"Verma, Sanjeev Kumar"https://zbmath.org/authors/?q=ai:verma.sanjeev-kumarSummary: We provide a novel approach based on \(\chi^2\) analysis that investigates regions in the neutrino parameter space where long baseline experiments cannot discriminate between the \((3 + 1)\nu\) oscillation scheme and the standard \(3\nu\) oscillation scheme. We call such region of parameter space ``the inaccessible region''. The inaccessible area of parameter space is so named because the minimal value of \(\chi^2\) remains almost unaltered even after inserting the sterile neutrino in the description of the neutrino oscillation data. Such inaccessible zones are discovered for the three long-baseline experiments (T2K, NOva and DUNE). Finally, we find the region of the sterile neutrino parameter space that remains inaccessible even after combining all three experiments. If sterile neutrinos exist in nature but their mixing parameters are in this range, these long-baseline experiments cannot prove their existence. The significance of our finding stems from the likelihood of sterile neutrinos existing in the accessible zone, which may be tested in these experiments in near future.