On the Djrbashian kernel of a Siegel domain

Barletta, Elisabetta; Dragomir, Sorin

Studia Mathematica, Tome 129 (1998), p. 47-63 / Harvested from The Polish Digital Mathematics Library

Résumé

We establish an inversion formula for the M. M. Djrbashian A. H. Karapetyan integral transform (cf. [6]) on the Siegel domain $Ω_{n} = ζ \in ℂ^{n} : ϱ (ζ) > 0$ , $ϱ (ζ) = I m (ζ_{1}) - {| ζ^{'} |}^{2}$ . 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 $V_{a}$ , $a \in B_{n}$ .

Publié le : 1998-01-01

Zbl 0903.32008

EUDML-ID : urn:eudml:doc:216459

@article{bwmeta1.element.bwnjournal-article-smv127i1p47bwm,
     author = {Elisabetta Barletta and Sorin Dragomir},
     title = {On the Djrbashian kernel of a Siegel domain},
     journal = {Studia Mathematica},
     volume = {129},
     year = {1998},
     pages = {47-63},
     zbl = {0903.32008},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv127i1p47bwm}
}

Barletta, Elisabetta; Dragomir, Sorin. On the Djrbashian kernel of a Siegel domain. Studia Mathematica, Tome 129 (1998) pp. 47-63. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv127i1p47bwm/

Bibliographie

[00000] [1] N. Aronszajn, La théorie des noyaux reproduisants et ses applications, Proc. Cambridge Philos. Soc. 39 (1943), 118-153. | Zbl 0061.26204

[00001] [2] S. Bergman, Über die Kernfunktion eines Bereiches und ihr Verhalten am Rande. I, J. Reine Angew. Math. 169 (1933), 1-42.

[00002] [3] S. Bergman, The Kernel Function and Conformal Mapping, Math. Surveys 5, Amer. Math. Soc., 1950. | Zbl 0040.19001

[00003] [4] R. R. Coifman and R. Rochberg, Representation theorems for holomorphic and harmonic functions in $L^{p}$ , Astérisque 77 (1980), 11-66. | Zbl 0472.46040

[00004] [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).

[00005] [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. | Zbl 0791.32004

[00006] [7] S. Dragomir, On weighted Bergman kernels of bounded domains, Studia Math., 108 (1994), 149-157. | Zbl 0818.32006

[00007] [8] K. Gawędzki, Fourier-like kernels in geometric quantization, Dissertationes Math. 125 (1976). | Zbl 0343.53024

[00008] [9] T. Genchev, Paley-Wiener type theorems for functions holomorphic in a half-plane, C. R. Acad. Bulgare Sci. 37 (1983), 141-144. | Zbl 0545.30030

[00009] [10] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York, 1978. | Zbl 0451.53038

[00010] [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. | Zbl 0422.53032

[00011] [12] S. Kobayashi, Geometry of bounded domains, Trans. Amer. Math. Soc. 92 (1959), 267-290. | Zbl 0136.07102

[00012] [13] S. G. Krantz, Function Theory of Several Complex Variables, Pure Appl. Math., Wiley, New York, 1982. | Zbl 0471.32008

[00013] [14] A. Lichnerowicz, Variétés complexes et tenseur de Bergman, Ann. Inst. Fourier (Grenoble) 15 (1965), 345-408. | Zbl 0134.05903

[00014] [15] T. Mazur, Canonical isometry on weighted Bergman spaces, Pacific J. Math. 136 (1989), 303-310. | Zbl 0677.46015

[00015] [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.

[00016] [17] T. Mazur and M. Skwarczyński, Spectral properties of holomorphic automorphisms with fixed point, Glasgow Math. J. 28 (1986), 25-30. | Zbl 0579.46017

[00017] [18] A. Odzijewicz, On reproducing kernels and quantization of states, Comm. Math. Phys. 114 (1988), 577-597. | Zbl 0645.53044

[00018] [19] Z. Pasternak-Winiarski, On the dependence of the reproducing kernel on the weight of integration, J. Funct. Anal. 94 (1990), 110-134. | Zbl 0739.46010

[00019] [20] Z. Pasternak-Winiarski, On weights which admit the reproducing kernel of Bergman type, Internat. J. Math. Math. Sci. 15 (1992), 1-14. | Zbl 0749.32019

[00020] [21] W. Rudin, Function Theory in the Unit Ball of $ℂ^{n}$ , Springer, New York, 1980.

[00021] [22] S. Saitoh, Hilbert spaces induced by Hilbert space valued functions, Proc. Amer. Math. Soc. 89 (1983), 74-78. | Zbl 0595.46026

[00022] [23] S. Saitoh, One approach to some general integral transforms and its applications, Integral Transforms and Special Functions 3 (1995), 49-84. | Zbl 0837.30002

[00023] [24] M. Skwarczyński, Biholomorphic invariants related to the Bergman function, Dissertationes Math. 173 (1980).

[00024] [25] M. Skwarczyński, Alternating projections between a strip and a half-plane, Math. Proc. Cambridge Philos. Soc. 102 (1987), 121-129. | Zbl 0625.30012