Homogeneously Polyanalytic Kernels on the Unit Ball and the Siegel Domain

Christian Rene Leal-Pacheco; Egor A. Maximenko; Gerardo Ramos-Vazquez

doi:10.1007/s11785-021-01145-z

Homogeneously Polyanalytic Kernels on the Unit Ball and the Siegel Domain

Christian Rene Leal-Pacheco, Egor A. Maximenko, Gerardo Ramos-Vazquez

Escuela Superior de Física y Matemáticas (ESFM)

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

We prove that the homogeneously polyanalytic functions of total order m, defined by the system of equations D¯(k1,…,kn)f=0 with k₁+ ⋯ + k_n= m, can be written as polynomials of total degree < m in variables z₁¯ , … , 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 Cⁿ and on the Siegel domain. For the one-dimensional case, analogous results were obtained by Koshelev (Akad. Nauk SSSR 232: 277–279, 1977), Pessoa (Complex Anal. Oper. Theory 8: 359–381, 2014), Hachadi and Youssfi (Complex Anal. Oper. Theory 13: 3457–3478, 2019).

Translated title of the contribution	Núcleos polianalíticos homogéneos en la bola unitaria y el dominio de Siegel
Original language	English
Article number	99
Journal	Complex Analysis and Operator Theory
Volume	15
Issue number	6
DOIs	https://doi.org/10.1007/s11785-021-01145-z
State	Published - Sep 2021

Keywords

Bergman space
Jacobi polynomial
Mean value property
Möbius transform
Polyanalytic function of several variables
Pseudohyperbolic distance
Reproducing kernel

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title = "Homogeneously Polyanalytic Kernels on the Unit Ball and the Siegel Domain",

abstract = "We prove that the homogeneously polyanalytic functions of total order m, defined by the system of equations D¯(k1,…,kn)f=0 with k1+ ⋯ + kn= m, can be written as polynomials of total degree < m in variables z1¯ , … , zn¯ , 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 Cn and on the Siegel domain. For the one-dimensional case, analogous results were obtained by Koshelev (Akad. Nauk SSSR 232: 277–279, 1977), Pessoa (Complex Anal. Oper. Theory 8: 359–381, 2014), Hachadi and Youssfi (Complex Anal. Oper. Theory 13: 3457–3478, 2019).",

keywords = "Bergman space, Jacobi polynomial, Mean value property, M{\"o}bius transform, Polyanalytic function of several variables, Pseudohyperbolic distance, Reproducing kernel",

author = "Leal-Pacheco, {Christian Rene} and Maximenko, {Egor A.} and Gerardo Ramos-Vazquez",

note = "Publisher Copyright: {\textcopyright} 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG.",

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AU - Leal-Pacheco, Christian Rene

AU - Maximenko, Egor A.

AU - Ramos-Vazquez, Gerardo

PY - 2021/9

Y1 - 2021/9

N2 - We prove that the homogeneously polyanalytic functions of total order m, defined by the system of equations D¯(k1,…,kn)f=0 with k1+ ⋯ + kn= m, can be written as polynomials of total degree < m in variables z1¯ , … , zn¯ , 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 Cn and on the Siegel domain. For the one-dimensional case, analogous results were obtained by Koshelev (Akad. Nauk SSSR 232: 277–279, 1977), Pessoa (Complex Anal. Oper. Theory 8: 359–381, 2014), Hachadi and Youssfi (Complex Anal. Oper. Theory 13: 3457–3478, 2019).

AB - We prove that the homogeneously polyanalytic functions of total order m, defined by the system of equations D¯(k1,…,kn)f=0 with k1+ ⋯ + kn= m, can be written as polynomials of total degree < m in variables z1¯ , … , zn¯ , 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 Cn and on the Siegel domain. For the one-dimensional case, analogous results were obtained by Koshelev (Akad. Nauk SSSR 232: 277–279, 1977), Pessoa (Complex Anal. Oper. Theory 8: 359–381, 2014), Hachadi and Youssfi (Complex Anal. Oper. Theory 13: 3457–3478, 2019).

KW - Bergman space

KW - Jacobi polynomial

KW - Mean value property

KW - Möbius transform

KW - Polyanalytic function of several variables

KW - Pseudohyperbolic distance

KW - Reproducing kernel

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DO - 10.1007/s11785-021-01145-z

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VL - 15

JO - Complex Analysis and Operator Theory

JF - Complex Analysis and Operator Theory

IS - 6

M1 - 99

ER -

Homogeneously Polyanalytic Kernels on the Unit Ball and the Siegel Domain

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Keywords

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