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1998 | 127 | 1 | 47-63
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On the Djrbashian kernel of a Siegel domain

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EN
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} $, $ϱ(ζ) = Im(ζ_1) - |ζ'|^2$. 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 $V_a$, $a ∈ B_n$.
Twórcy
  • Dipartimento di Matematica, Università della Basilicata, Via N. Sauro 85, 85100 Potenza, Italy , barletta@unibas.it
  • Dipartimento di Matematica, Università della Basilicata, Via N. Sauro 85, 85100 Potenza, Italy , dragomir@unibas.it
Bibliografia
  • [1] N. Aronszajn, La théorie des noyaux reproduisants et ses applications, Proc. Cambridge Philos. Soc. 39 (1943), 118-153.
  • [2] S. Bergman, Über die Kernfunktion eines Bereiches und ihr Verhalten am Rande. I, J. Reine Angew. Math. 169 (1933), 1-42.
  • [3] S. Bergman, The Kernel Function and Conformal Mapping, Math. Surveys 5, Amer. Math. Soc., 1950.
  • [4] R. R. Coifman and R. Rochberg, Representation theorems for holomorphic and harmonic functions in $L^p$, Astérisque 77 (1980), 11-66.
  • [5] M. M. Djrbashian, Interpolation and spectral expansions associated with differential operators of fractional order, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 19 (1984), 81-181 (in Russian).
  • [6] M. M. Djrbashian and A. H. Karapetyan, Integral representations for some classes of functions holomorphic in a Siegel domain, J. Math. Anal. Appl. 179 (1993), 91-109.
  • [7] S. Dragomir, On weighted Bergman kernels of bounded domains, Studia Math., 108 (1994), 149-157.
  • [8] K. Gawędzki, Fourier-like kernels in geometric quantization, Dissertationes Math. 125 (1976).
  • [9] T. Genchev, Paley-Wiener type theorems for functions holomorphic in a half-plane, C. R. Acad. Bulgare Sci. 37 (1983), 141-144.
  • [10] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York, 1978.
  • [11] P. F. Klembeck, Kähler metrics of negative curvature, the Bergman metric near the boundary, and the Kobayashi metric on smooth bounded pseudoconvex sets, Indiana Univ. Math. J. (2) 27 (1978), 275-282.
  • [12] S. Kobayashi, Geometry of bounded domains, Trans. Amer. Math. Soc. 92 (1959), 267-290.
  • [13] S. G. Krantz, Function Theory of Several Complex Variables, Pure Appl. Math., Wiley, New York, 1982.
  • [14] A. Lichnerowicz, Variétés complexes et tenseur de Bergman, Ann. Inst. Fourier (Grenoble) 15 (1965), 345-408.
  • [15] T. Mazur, Canonical isometry on weighted Bergman spaces, Pacific J. Math. 136 (1989), 303-310.
  • [16] T. Mazur, On the complex manifolds of Bergman type, in: Classical Analysis, Proc. 6th Symposium (23-29 September 1991, Poland), World Scientific, 1993, 132-138.
  • [17] T. Mazur and M. Skwarczyński, Spectral properties of holomorphic automorphisms with fixed point, Glasgow Math. J. 28 (1986), 25-30.
  • [18] A. Odzijewicz, On reproducing kernels and quantization of states, Comm. Math. Phys. 114 (1988), 577-597.
  • [19] Z. Pasternak-Winiarski, On the dependence of the reproducing kernel on the weight of integration, J. Funct. Anal. 94 (1990), 110-134.
  • [20] Z. Pasternak-Winiarski, On weights which admit the reproducing kernel of Bergman type, Internat. J. Math. Math. Sci. 15 (1992), 1-14.
  • [21] W. Rudin, Function Theory in the Unit Ball of $ℂ^n$, Springer, New York, 1980.
  • [22] S. Saitoh, Hilbert spaces induced by Hilbert space valued functions, Proc. Amer. Math. Soc. 89 (1983), 74-78.
  • [23] S. Saitoh, One approach to some general integral transforms and its applications, Integral Transforms and Special Functions 3 (1995), 49-84.
  • [24] M. Skwarczyński, Biholomorphic invariants related to the Bergman function, Dissertationes Math. 173 (1980).
  • [25] M. Skwarczyński, Alternating projections between a strip and a half-plane, Math. Proc. Cambridge Philos. Soc. 102 (1987), 121-129.
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