Approximation by polyanalytic polynomials

Fedorovskiy, Konstantin Yurievich

doi:10.20948/mono-2016-fedorovsky

Cited by 4 publications

(1 citation statement)

References 33 publications

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“…Polyanalytic functions, have been attracted attention of many mathematicians since the beginning of the 20th century. See some of their properties, applications, and history in [1,2,5,12,14].…”

Section: Introductionmentioning

confidence: 99%

Homogeneously polyanalytic kernels on the unit ball and the Siegel domain

Leal-Pacheco,

Maximenko,

Ramos-Vazquez

2021

Preprint

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We prove that the homogeneously polyanalytic functions of total order m, defined by the system of equations Dbe written as polynomials of total degree < m in variables z 1 , . . . , z n , with some analytic coefficients. We establish a weighted mean value property for such functions, using a reproducing property of Jacobi polynomials. After that, we give a general recipe to transform a reproducing kernel by a weighted change of variables.Applying these tools, we compute the reproducing kernel of the Bergman space of homogeneously polyanalytic functions on the unit ball in C n and on the Siegel domain. For the one-dimensional case, analogous results were obtained by Koshelev (1977), Pessoa (2014), Hachadi and Youssfi (2019).

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Section: Introductionmentioning

confidence: 99%

Homogeneously polyanalytic kernels on the unit ball and the Siegel domain

Leal-Pacheco,

Maximenko,

Ramos-Vazquez

2021

Preprint

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On the Polyanalytic and Anti-Polyanalytic Function Spaces

Vasilevski

2022

J Math Sci

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Let D be either or ℂ , and let J be either [0, 1] or ℝ + , so that D = J × , where is the unit circle in ℂ . We set H for any weighted Hilbert space L 2 (D, d ) , with the probability measure d (z) = (|z|)dA(z) , where dA(z) = 1 dxdy , z = x + iy , and whose radial weight function ∶ D → ℝ + is such that the linear span of the monomials z p z q , for all p, q ∈ ℤ + , is dense in H . We develop a unified approach to the characterization of the polyanalytic and anti-polyanalytic function spaces on such H . We do this in two different ways. The first one is based on the use of the orthonormal basis in H , while the second one is based on the use of the operators z and zI , and thus does not depend on a specific space H in question. Several examples illustrating and developing the results for specific spaces H are given.

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