TY - JOUR
T1 - Homogeneously Polyanalytic Kernels on the Unit Ball and the Siegel Domain
AU - Leal-Pacheco, Christian Rene
AU - Maximenko, Egor A.
AU - Ramos-Vazquez, Gerardo
N1 - Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
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
UR - http://www.scopus.com/inward/record.url?scp=85112006857&partnerID=8YFLogxK
U2 - 10.1007/s11785-021-01145-z
DO - 10.1007/s11785-021-01145-z
M3 - Artículo
AN - SCOPUS:85112006857
SN - 1661-8254
VL - 15
JO - Complex Analysis and Operator Theory
JF - Complex Analysis and Operator Theory
IS - 6
M1 - 99
ER -