On the Djrbashian kernel of a Siegel domain

Elisabetta Barletta; Sorin Dragomir

Studia Mathematica (1998)

  • Volume: 127, Issue: 1, page 47-63
  • ISSN: 0039-3223

Abstract

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We establish an inversion formula for the M. M. Djrbashian A. H. Karapetyan integral transform (cf. [6]) on the Siegel domain Ω n = ζ ∈ ℂ n : ϱ ( ζ ) > 0 , ϱ ( ζ ) = I m ( ζ 1 ) - | ζ ' | 2 . We build a family of Kähler metrics of constant holomorphic curvature whose potentials are the ϱ α -Bergman kernels, α > -1, (in the sense of Z. Pasternak-Winiarski [20] of Ω n . We build an anti-holomorphic embedding of Ω n in the complex projective Hilbert space ℂ ℙ ( H α 2 ( Ω n ) ) and study (in connection with work by A. Odzijewicz [18] the corresponding transition probability amplitudes. The Genchev transform (cf. [9]) is shown to be well defined on L 2 ( Ω , ϱ α ) , for any strip Ω ⊂ ℂ, and applied in a problem of approximation by holomorphic functions. Building on work by T. Mazur (cf. [15]) we prove the existence of a complete orthonormal system in H α 2 ( Ω n ) consisting of eigenfunctions of a certain explicitly defined operator V a , a ∈ B n .

How to cite

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Barletta, Elisabetta, and Dragomir, Sorin. "On the Djrbashian kernel of a Siegel domain." Studia Mathematica 127.1 (1998): 47-63. <http://eudml.org/doc/216459>.

@article{Barletta1998,
abstract = {We establish an inversion formula for the M. M. Djrbashian A. H. Karapetyan integral transform (cf. [6]) on the Siegel domain $Ω_n = \{ζ ∈ ℂ^n : ϱ (ζ) >0\} $, $ϱ(ζ) = Im(ζ_1) - |ζ^\{\prime \}|^2$. We build a family of Kähler metrics of constant holomorphic curvature whose potentials are the $ϱ^α$-Bergman kernels, α > -1, (in the sense of Z. Pasternak-Winiarski [20] of $Ω_n$. We build an anti-holomorphic embedding of $Ω_n$ in the complex projective Hilbert space $ℂℙ(H^2_α(Ω_n))$ and study (in connection with work by A. Odzijewicz [18] the corresponding transition probability amplitudes. The Genchev transform (cf. [9]) is shown to be well defined on $L^2(Ω, ϱ^α)$, for any strip Ω ⊂ ℂ, and applied in a problem of approximation by holomorphic functions. Building on work by T. Mazur (cf. [15]) we prove the existence of a complete orthonormal system in $H^2_α(Ω_n)$ consisting of eigenfunctions of a certain explicitly defined operator $V_a$, $a ∈ B_n$.},
author = {Barletta, Elisabetta, Dragomir, Sorin},
journal = {Studia Mathematica},
keywords = {γ-Bergman kernel; reproducing kernel Hilbert space; Djrbashian kernel; transition probability amplitude; Genchev transform; Bergman kernel},
language = {eng},
number = {1},
pages = {47-63},
title = {On the Djrbashian kernel of a Siegel domain},
url = {http://eudml.org/doc/216459},
volume = {127},
year = {1998},
}

TY - JOUR
AU - Barletta, Elisabetta
AU - Dragomir, Sorin
TI - On the Djrbashian kernel of a Siegel domain
JO - Studia Mathematica
PY - 1998
VL - 127
IS - 1
SP - 47
EP - 63
AB - We establish an inversion formula for the M. M. Djrbashian A. H. Karapetyan integral transform (cf. [6]) on the Siegel domain $Ω_n = {ζ ∈ ℂ^n : ϱ (ζ) >0} $, $ϱ(ζ) = Im(ζ_1) - |ζ^{\prime }|^2$. We build a family of Kähler metrics of constant holomorphic curvature whose potentials are the $ϱ^α$-Bergman kernels, α > -1, (in the sense of Z. Pasternak-Winiarski [20] of $Ω_n$. We build an anti-holomorphic embedding of $Ω_n$ in the complex projective Hilbert space $ℂℙ(H^2_α(Ω_n))$ and study (in connection with work by A. Odzijewicz [18] the corresponding transition probability amplitudes. The Genchev transform (cf. [9]) is shown to be well defined on $L^2(Ω, ϱ^α)$, for any strip Ω ⊂ ℂ, and applied in a problem of approximation by holomorphic functions. Building on work by T. Mazur (cf. [15]) we prove the existence of a complete orthonormal system in $H^2_α(Ω_n)$ consisting of eigenfunctions of a certain explicitly defined operator $V_a$, $a ∈ B_n$.
LA - eng
KW - γ-Bergman kernel; reproducing kernel Hilbert space; Djrbashian kernel; transition probability amplitude; Genchev transform; Bergman kernel
UR - http://eudml.org/doc/216459
ER -

References

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