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polyanalytic Toeplitz Operators: Isomorphisms, Symbolic Calculus and Approximation of Weyl Operators

JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS 2021年第3期27卷 1-34页

作者： Keller, Johannes Luef, Franz NTNU Trondheim Dept Math Sci N-7041 Trondheim Norway

We discuss an extension of Toeplitz quantization based on polyanalytic functions. We derive isomorphism theorem for polyanalytic Toeplitz operators between weighted Sobolev-Fock spaces of polyanalytic functions, which are images of modulation spaces under polyanalytic Bargmann transforms. This generalizes well-known results from the analytic setting. Finally, we derive an asymptotic symbol calculus and present an asymptotic expansion of complex Weyl operators in terms of polyanalytic Toeplitz operators.

关键词： Bargmann transform polyanalytic functions Toeplitz quantization Sobolev-Fock space Symbolic calculus Semiclassical approximation

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A special orthogonal complement basis for holomorphic-Hermite functions and associated 1d-and 2d-fractional Fourier transforms

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INTEGRAL TRANSFORMS AND SPECIAL functions 2020年第6期31卷 471-486页

作者： Benahmadi, Abdelhadi Ghanmi, Allal El Ainin, Mohammed Souid Mohammed V Univ Rabat CeReMAR Fac SciAGA Lab MIA SIDept Math Anal PDE & Specral Geometry POB 1014 Rabat Morocco Ibn Zohr Univ Fac Law Econ & Social Sci Agadir Morocco

In 1990, van Eijndhoven and Meyers provide a special orthonormal basis for the Bargmann Hilbert space consisting of holomorphic Hermite functions. Then it was be natural to look for its orthogonal complement in the underlying -Hilbert space. In this paper, we describe the orthogonal complement of this Hilbert space. More precisely, a polyanalytic orthonormal basis is given and the explicit expressions of the corresponding reproducing kernel functions and Segal-Bargmann integral transforms are provided. The obtained basis are then used to provide a non-trivial - and -fractional like-Fourier transforms.

关键词： Generalized Bargmann spaces polyanalytic functions reproducing kernels holomorphic Hermite polynomials polyanalytic Hermite polynomials Segal-Bargmann transform fractional like-Fourier transform

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Carathéodory Sets in the Plane

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2024年

作者： Joan Josep Carmona Konstantin Fedorovskiy

ISBN: (数字)9783985475728

ISBN: (纸本)9783985470723

This work is devoted to the class of sets in the complex plane which nowadays are known as Carathéodory sets, more precisely speaking, as Carathéodory domains and Carathéodory compact sets. These sets naturally arose many times in various research areas in Real, Complex and Functional Analysis and in the Theory of Partial Differential Equations. For instance, the concept of a Carathéodory set plays a significant role in such topical themes as approximation in the complex plane, the theory of conformal mappings, boundary value problems for elliptic partial differential equations, etc. The first appearance of Carathéodory domains in the mathematical literature (of course, without the special name at that moment) was at the beginning of the 20th century, when C. Carathéodory published his famous series of papers about boundary behavior of conformal mappings. The next breakthrough result which was obtained with the essential help of this concept is the Walsh–Lebesgue criterion for uniform approximation of functions by harmonic polynomials on plane compacta (1929). Up to now the studies of Carathéodory domains and Carathéodory compact sets remains a topical field of contemporary analysis and a number of important results were recently obtained in this direction. Among them one ought to mention the results about polyanalytic polynomial approximation, where the class of Carathéodory compact sets was one of the crucial tools, and the results about boundary behavior of conformal mappings from the unit disk onto Carathéodory domains. Our aim in the present paper is to give a survey on known results related with Carathéodory sets and to present several new results concerning the matter. Starting with the classical works of Carathéodory, Farrell, Walsh, and passing through the history of Complex Analysis of the 20th century, we come to recently obtained results, and to our contribution to the theory.

关键词： Carathéodory domain Carathéodory compact set conformal mapping uniform approximation pointwise approximation Lp -approximation complex polynomials complex rational functions harmonic functions polyanalytic functions

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Fixed points of the Berezin transform of multidimensional polyanalytic Fock spaces

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2020年第2期481卷

作者： Casseli, Irene Aix Marseille Univ I2M UMR CNRS 7373 39 Rue F Joliot Curie F-13453 Marseille 13 France

We study the fixed points of the Berezin transform in polyanalytic Fock spaces of C-d, d is an element of N*. We show that an L-p function, p is an element of [1, + infinity], with respect to the Lebesgue measure is invariant under this transformation if and only if it is harmonic. From this we deduce that the only bounded fixed points of the Berezin transform of polyanalytic Fock spaces are constant functions. (C) 2019 Elsevier Inc. All rights reserved.

关键词： polyanalytic functions Berezin transform Fock spaces Laguerrc polynomials

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Donoho-Logan large sieve principles for modulation and polyanalytic Fock spaces

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BULLETIN DES SCIENCES MATHEMATIQUES 2021年 171卷

作者： Abreu, Luis Daniel Speckbacher, Michael Austrian Acad Sci Acoust Res Inst Wohllebengasse 12-14 A-1040 Vienna Austria

We obtain estimates for the L-p-norm of the short-time Fourier transform (STFT) for functions in modulation spaces, providing information about the concentration on a given subset of R-2, leading to deterministic guarantees for perfect reconstruction using convex optimization methods. More precisely, we obtain large sieve inequalities of the Donoho-Logan type, but instead of localizing the signals in regions T x W of the time-frequency plane using the Fourier transform to intertwine time and frequency, we localize the representation of the signals in terms of the short-time Fourier transform in sets Delta with arbitrary geometry. At the technical level, since there is no proper analogue of Beurling's extremal function in the STFT setting, we introduce a new method, which rests on a combination of an argument similar to Schur's test with an extension of Seip's local reproducing formula to general Hermite windows. When the windows are Hermite functions, we obtain local reproducing formulas for polyanalytic Fock spaces which lead to explicit large sieve constant estimates and, as a byproduct, to a reconstruction formula for f is an element of L-2 (R) from its STFT values on arbitrary discs. (C) 2021 Published by Elsevier Masson SAS.

关键词： Large sieve Gabor analysis polyanalytic functions Higher Landau levels Modulation spaces Signal recovering

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On a Novel Class of polyanalytic Hermite Polynomials

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RESULTS IN MATHEMATICS 2019年第4期74卷 1-23页

作者： Benahmadi, Abdelhadi Ghanmi, Allal Mohammed V Univ Fac Sci Dept Math Anal PDE & Spectral Geometry POB 1014 Rabat Morocco

We discuss some algebraic and analytic properties of a general class of orthogonal polyanalytic polynomials, including their operational formulas, recurrence relations, generating functions, integral representations and different orthogonality identities. We establish their connection and rule in describing the L-2-spectral theory of some special second order differential operators of Laplacian type acting on the L-2-Gaussian Hilbert space on the whole complex plane. We will also show their importance in the theory of the so-called rank-one automorphic functions on the complex plane. In fact, a variant subclass leads to an orthogonal basis of the corresponding L-2-Gaussian Hilbert space on the strip C/Z.

关键词： Holomorphic Hermite polynomial polyanalytic complex Hermite polynomial generating function orthogonality relation integral representation polyanalytic functions rank-one autmorphic theta functions

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A necessary condition for weak maximum modulus sets of 2-analytic functions

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COLLECTANEA MATHEMATICA 2018年第2期69卷 173-180页

作者： Daghighi, Abtin Linkoping Univ S-58183 Linkoping Sweden

Let Omega subset of C be a domain and let f (z) = a(z) + (z) over barb(z), where a, b are holomorphic for z is an element of Omega. Denote by. the set of points in Omega at which vertical bar f vertical bar| attains weak local maximum and denote by Sigma the set of points in Omega at which vertical bar f vertical bar attains strict local maximum. We prove that for each p is an element of Lambda \ Sigma, Furthermore, if there is a real analytic curve kappa : I Lambda\ Sigma ( where I is an open real interval), if a, b are complex polynomials, and if f o kappa has a complex polynomial extension, then either f is constant or kappa has constant curvature.

关键词： polyanalytic functions q-Analytic functions Peak sets Maximum modulus sets

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ON NEVANLINNA DOMAINS WITH FRACTAL BOUNDARIES

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ST PETERSBURG MATHEMATICAL JOURNAL 2018年第5期29卷 777-791页

作者： Mazalov, M. Ya Natl Res Univ Moscow Energy Inst Smolensk Branch Smolensk Russia Moscow NE Bauman State Tech Univ Moscow Russia

A positive answer is given to the question on the existence of a Nevanlinna contour of Hausdorff dimension exceeding 1, posed by K. Yu. Fedorovskii in 2001. In particular, it is shown that this dimension may exceed 3/2.

关键词： Nevanlinna contours and domains conformal mapping univalent functions Blaschke condition polyanalytic functions fractals

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The Plemelj-Privalov theorem in polyanalytic function theory

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2018年第2期463卷 517-533页

作者： De la Cruz Toranzo, Lianet Abreu Blaya, Ricardo Bory Reyes, Juan Univ Holguin Fac Informat & Matemat Holguin 80100 Cuba Inst Politecn Nacl SEPI ESIME ZAC Mexico City 07738 Cd Mx Mexico

In this paper we prove that the higher order Lipschitz classes behave invariant under the action of a singular integral operator naturally arising in polyanalytic function theory. This result provides a generalization of the well-known theorem by Joseph Plemelj [16] and Ivan Privalov [17]. (C) 2018 Elsevier Inc. All rights reserved.

关键词： Singular integral operator Lipschitz classes polyanalytic functions

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Level Sets of Certain Subclasses of α-Analytic functions

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Journal of Partial Differential Equations 2017年第4期30卷 281-298页

作者： DAGHIGHI Abtin WIKSTROM Frank Mid Sweden University Holmgatan 10 SE-851 70 Sundsvall Sweden Center for Mathematical Sciences Lund University Box 118 SE-22100 Lund Sweden

For an open set V C Cn, denote by Mα （V） the family of α-analytic functions that obey a boundary maximum modulus principle. We prove that, on a bounded ＂harmonically fat＂ domain Ω C Cn, a function f ∈M a（Ω／f-1（0）） automatically sat- isfies f ∈M a（Ω）, if it is Caj-1smooth in the z/variable, α ∈ Zn＋ up to the boundary. For a submanifold U C Cn, denote by ma（U）, the set of functions locally approximable by α-analytic functions where each approximating member and its reciprocal （off the singularities） obey the boundary maximum modulus principle. We prove, that for a C3-smooth hypersurface, Ω, a member of ma （Ω）, cannot have constant modulus near a point where the Levi form has a positive eigenvalue, unless it is there the trace of a polyanalytic function of a simple form. The result can be partially generalized to C4-smooth submanifolds of higher codimension, at least near points with a Levi cone condition.

关键词： polyanalytic functions q-analytic functions zero sets level sets a-analytic functions.

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