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Bohr operator on operator-valued polyanalytic functions on simply connected domains

CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES 2023年第4期66卷 1411-1422页

作者： Allu, Vasudevarao Halder, Himadri Indian Inst Technol Bhubaneswar Sch Basic Sci Bhubaneswar 752050 India Indian Inst Technol Math Dept Mumbai 400076 India

this article, we study the Bohr operator for the operator-valued subordination class S( f) consisting of holomorphic functions subordinate to f in the unit disk D := {z. C : |z| < 1}, where f : D. B(H) is holomorphic and B(H) is the algebra of bounded linear operators on a complex Hilbert space H. We establish several subordination results, which can be viewed as the analogs of a couple of interesting subordination results from scalar-valued settings. We also obtain a von Neumann-type inequality for the class of analytic self-mappings of the unit disk D which fix the origin. Furthermore, we extensively study Bohr inequalities for operator-valued polyanalytic functions in certain proper simply connected domains in C. We obtain Bohr radius for the operator-valued polyanalytic functions of the form F(z) = Sigma (p-1) (l=0) (-1) (z), where f0 is subordinate to an operatorvalued convex biholomorphic function, and operator-valued starlike biholomorphic function in the unit disk D.

关键词： Banach algebra von Neumann inequality polyanalytic functions Bohr operator simply connected domains

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Local Maximum Modulus Property for polyanalytic functions

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COMPLEX ANALYSIS AND OPERATOR THEORY 2016年第2期10卷 401-408页

作者： Daghighi, Abtin Krantz, Steven G. Linkoping Univ SE-58183 Linkoping Sweden Washington Univ Dept Math St Louis MO 63130 USA

Let be an open subset and let be a space of functions defined on . is said to have the local maximum modulus property if: for every and for every sufficiently small domain with it holds true that where denotes the set of points at which attains strict local maximum. This property fails for We verify it however for the set of complex-valued functions whose real and imaginary parts are real analytic. We show by example that the property cannot be improved upon whenever is the set of n-analytic functions on , in the sense that locality cannot be removed as a condition and independently cannot be removed from the conclusion.

关键词： polyanalytic functions n-analytic functions Boundary maximum modulus principle Local maximum modulus property

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BOUNDARY VALUE PROBLEMS OF CONJUGATE AND GENERALIZED K-HOLOMORPHIC functions IN C^(2)

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Acta Mathematica Scientia 2024年第5期44卷 1837-1852页

作者： Yanyan CUI Chaojun WANG Yonghong XIE Yuying QIAO College of Mathematics and Statistics Zhoukou Normal UniversityZhoukou466001China College of Mathematics and Information Science Hebei Normal UniversityShijiazhuang050024China

In this paper,conjugate k-holomorphic functions and generalized k-holomorphic functions are defined in the two-dimensional complex space,and the corresponding Riemann boundary value problems and the inverse problems are discussed on generalized *** the characteristics of the corresponding functions and boundary properties of the Cauchy type singular integral operators with conjugate k-holomorphic kernels,the general solutions and special solutions of the corresponding boundary value problems are studied in a detailed fashion,and the integral expressions of the solutions are obtained.

关键词： several complex variables k-holomorphic functions partial difference equations polyanalytic functions

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Weighted Bergman kernels for nearly holomorphic functions on bounded symmetric domains

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JOURNAL OF FUNCTIONAL ANALYSIS 2024年第2期286卷

作者： Englis, Miroslav Youssfi, El-Hassan Zhang, Genkai Silesian Univ Opava Math Inst Na Rybnicku 1 Opava 74601 Czech Republic Math Inst Zitna 25 Prague 1 Czech Republic Aix Marseille Univ UMR CNRS I2M 7373 39 Rue F Juliot Curie F-13453 Aix En Provence 13 France Gothenburg Univ Chalmers Univ Technol Math Sci SE-41296 Gothenburg Sweden

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 "invariantlypolyanalytic" 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-Op dam hyper geometric 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 symmet ric spaces, where explicit formulas in terms of multivariable Jacobi polynomials are given.(c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).

关键词： Nearly holomorphic functions polyanalytic functions Bergman kernel Bounded symmetric domain

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Local Maxima of White Noise Spectrograms and Gaussian Entire functions

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JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS 2022年第6期28卷 1-28页

作者： Abreu, Luis Daniel Univ Vienna Fac Math NuHAG Oskar Morgenstern Pl 1 A-1090 Vienna Austria Austrian Acad Sci Acoust Res Inst Wohllebengasse 12-14 A-1040 Vienna Austria

We confirm Flandrin's prediction for the expected average of local maxima of spectrograms of complex white noise with Gaussian windows (Gaussian spectrograms or, equivalently, modulus of weighted Gaussian Entire functions), a consequence of the conjectured double honeycomb mean model for the network of zeros and local maxima, where the area of local maxima centered hexagons is three times larger than the area of zero centered hexagons. More precisely, we show that Gaussian spectrograms, normalized such that their expected density of zeros is 1, have an expected density of 5/3 critical points, among those 1/3 are local maxima, and 4/3 saddle points, and compute the distributions of ordinate values (heights) for spectrogram local extrema. This is done by first writing the spectrograms in terms of Gaussian Entire functions (GEFs). The extrema are considered under the translation invariant derivative of the Fock space (which in this case coincides with the Chern connection from complex differential geometry). We also observe that the critical points of a GEF are precisely the zeros of a Gaussian random function in the first higher Landau level. We discuss natural extensions of these Gaussian random functions: Gaussian Weyl-Heisenberg functions and Gaussian bi-entire functions. The paper also reviews recent results on the applications of white noise spectrograms, connections between several developments, and is partially intended as a pedestrian introduction to the topic.

关键词： Local maxima White noise Spectrograms Gaussian entire functions Landau levels polyanalytic functions

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Properties of a polyanalytic functional calculus on the S-spectrum

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MATHEMATISCHE NACHRICHTEN 2023年第11期296卷 5190-5226页

作者： De Martino, Antonino Pinton, Stefano Politecn Milan Dipartimento Matemat Via Bonardi 9 I-20133 Milan Italy

The Fueter mapping theorem gives a constructive way to extend holomorphic functions of one complex variable to monogenic functions, that is, null solutions of the generalized Cauchy-Riemann operator in R-4, denoted by D. This theorem is divided in two steps. In the first step, a holomorphic function is extended to a slice hyperholomorphic function. The Cauchy formula for this type of functions is the starting point of the S-functional calculus. In the second step, a monogenic function is obtained by applying the Laplace operator in four real variables, namely, Delta, to a slice hyperholomorphic function. The polyanalytic functional calculus, that we study in this paper, is based on the factorization of Delta=DD. Instead of applying directly the Laplace operator to a slice hyperholomorphic function, we apply first the operator D and we get a polyanalytic function of order 2, that is, a function that belongs to the kernel of D-2. We can represent this type of functions in an integral form and then we can define the polyanalytic functional calculus on S-spectrum. The main goal of this paper is to show the principal properties of this functional calculus. In particular, we study a resolvent equation suitable for proving a product rule and generate the Riesz projectors.

关键词： F-functional calculus P-2-functional calculus polyanalytic functions Q-functional calculus resolvent equation Riesz projectors S-spectrum

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A polyanalytic FUNCTIONAL CALCULUS OF ORDER 2 ON THE S-SPECTRUM

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PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 2023年第6期151卷 2471-2488页

作者： De Martino, Antonino Pinton, Stefano Politecn Milan Dipartimento Matemat Via E Bonardi 9 I-20133 Milan Italy

. The Fueter theorem provides a two step procedure to build an axially monogenic function, i.e. a null-solution of the Cauchy-Riemann operator in R4, denoted by D. In the first step a holomorphic function is extended to a slice hyperholomorphic function, by means of the so-called slice operator. In the second step a monogenic function is built by applying the Laplace operator Delta in four real variables to the slice hyperholomorphic function. In this paper we use the factorization of the Laplace operator, i.e. Delta = DD to split the previous procedure. From this splitting we get a class of functions that lies between the set of slice hyperholomorphic functions and the set of axially monogenic functions: the set of axially polyanalytic functions of order 2, i.e. null-solutions of D2. We show an integral representation formula for this kind of functions. The formula obtained is fundamental to define the associated functional calculus on the S-spectrum.

关键词： polyanalytic functions quaternions functional calculus S-spectrum

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Hormander's L²-Method, <(?)over bar>-Problem and polyanalytic Function Theory in One Complex Variable

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COMPLEX ANALYSIS AND OPERATOR THEORY 2023年第3期17卷 1-25页

作者： Alpay, Daniel Colombo, Fabrizio Diki, Kamal Sabadini, Irene Struppa, Daniele C. Chapman Univ Schmid Coll Sci & Technol 1 Univ Drive Orange CA 92866 USA Politecn Milan Dipartimento Matemat Via E Bonardi 9 I-20133 Milan Italy Chapman Univ Donald Bren Presidential Chair Math 1 Univ Drive Orange CA 92866 USA

In this paper we consider the classical (?) over bar -problem in the case of one complex variable both for analytic and polyanalytic data. We apply the decomposition property of poly-analytic functions in order to construct particular solutions of this problem and obtain new Hormander type estimates using suitable powers of the Cauchy-Riemann oper-ator. We also compute particular solutions of the (?) over bar -problem for specific polyanalytic data such as the Ito complex Hermite polynomials and polyanalytic Fock kernels.

关键词： Hormander's estimate functions of exponential type Fock space polyanalytic functions (?)over-bar-problem A(p) spaces

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polyanalytic Hardy decomposition of higher order Lipschitz functions

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2021年第2期493卷

作者： Abreu Blaya, Ricardo De la Cruz Toranzo, Lianet Univ Autonoma Guerrero Fac Matemat Chilpancingo Mexico Univ Holguin Fac Informat & Matemat Holguin Cuba

This paper is concerned with the problem of decomposing a higher order Lipschitz function on a closed Jordan curve Gamma into a sum of two polyanalytic functions in each open domain defined by Gamma. Our basic tools are the Hardy projections related to a singular integral operator arising in polyanalytic function theory, which, as it is proved here, represents an involution operator on the higher order Lipschitz classes. Our result generalizes the classical Hardy decomposition of Holder continuous functions on the boundary of a domain. (C) 2020 Elsevier Inc. All rights reserved.

关键词： polyanalytic functions Higher order Lipschitz classes Hardy projections

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Sampling Trajectories for the Short-Time Fourier Transform

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JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS 2022年第6期28卷 1-23页

作者： Speckbacher, Michael Univ Vienna Fac Math Oskar Morgenstern Pl 1 A-1090 Vienna Austria

We study the problem of stable reconstruction of the short-time Fourier transform from samples taken from trajectories in R-2. We first investigate the interplay between relative density of the trajectory and the reconstruction property. Later, we consider spiraling curves, a special class of trajectories, and connect sampling and uniqueness properties of these sets. Moreover, we show that for window functions given by a linear combination of Hermite functions, it is indeed possible to stably reconstruct from samples on some particular natural choices of spiraling curves.

关键词： Gabor analysis Sampling theory Mobile sampling polyanalytic functions

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