AN ANALOGUE OF BREMERMANN’S THEOREM ON FINDING THE BERGMAN KERNEL FOR THE CARTESIAN PRODUCT OF THE CLASSICAL DOMAINS AND
Keywords:
Bergman’s kernel, classical domains, homogeneous domain, symmetric domain, Jacobian, orthonormal system
Abstract
In this paper, an analogue of Bremermann’s theorem on finding the Bergman kernel is obtained for the Cartesian product of classical domains. For this purpose, the groups of automorphisms of the considered domains are used, i.e., the Bergman kernels are constructed for the Cartesian product of classical domains, without applying complete orthonormal systems.
References
1. Shabat B. V. Introduction to Complex Analysis. Part II. Functions of Several Variables, Moscow: Nauka, Physical and mathematical literature, 1985, 464 p. (in Russian).
2. Aizenberg L. A., Yuzhakov A. P. Integral Representations and Residues in Multidimensional Complex Analysis, Novosibirsk: Nauka, 335 pp. (1979) (in Russian). Engl. transl.: Transl. Math. Monogr. Vol. 58, Providence, 283 pp. (1983).
3. Khudayberganov G., Kytmanov A. M., Shaimkulov B. A. Analysis in matrix domains. Monograph. Krasnoyarsk, Siberian Federal University, 297 p. (2017) (in Russian).
4. Hua Luogeng. Harmonic analysis of functions of several complex variables in classical domains, Inostr. Lit., M., 1959 (in Russian).
5. Khenkin, G. M. The method of integral representations in complex analysis, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr., 7, VINITI, Moscow, 1985, 23–124 (in Russian).
6. Kytmanov A. M., Nikitina T. N. Analogs of Carleman’s formula for classical domains, Mat. Zametki, 45:3 (1989), 87–93; Math. Notes, 45:3 (1989), 243–248.
7. Khudayberganov G., Abdullayev J. Sh. Relationship between the Kernels Bergman and Cauchy-Szegö in the domains and . J. Sib. Fed. Univ. Math. Phys., 13:5, 559–567 (2020).
8. Khudayberganov G., Rakhmonov U. S. The Bergman and Cauchy–Szegö kernels for matrix ball of the second type, J. Sib. Fed. Univ. Math. Phys., 7:3 (2014), 305–310.
9. Khudayberganov G., Rakhmonov U. S. Carleman Formula for Matrix Ball of the Third Type, Algebra, Complex Analysis, and Pluripotential Theory. USUZCAMP 2017. Springer Proceedings in Mathematics & Statistics, vol. 264, pp. 101–108, Springer, Cham. (2017).
10. Gindikin S. G. Analysis in homogeneous domains, Uspekhi Mat. Nauk, 19:4 (118) (1964), 3–92; Russian Math. Surveys, 19:4 (1964), 1–89.
11. Khudayberganov G., Rakhmonov U. S., Matyakubov Z. Q. Integral formulas for some matrix domains, Topics in Several Complex Variables, AMS, 2016, 89–95.
12. Myslivets S. G. Construction of Szegö and Poisson kernels in convex domains, J. Sib. Fed.Univ. Math. Phys., 2018, Volume 11, Issue 6, 792–795.
13. Rakhmonov U. S., Abdullayev J. Sh. On volumes of matrix ball of third type and generalized Lie balls, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 29:4 (2019), 548–557.
14. Myslivets S. G. On the Szegö and Poisson kernels in the convex domains in , Izv. Vyssh. Uchebn. Zaved. Mat., 2019, no. 1, 42–48.
15. Khudayberganov G., Khalknazarov A. M., Abdullayev J. Sh. Laplace and Hua Luogeng operators, Izv. Vyssh. Uchebn. Zaved. Mat., 2020, no. 3, 74–79.
16. Cartan E. Sur les domaines bornes homogenes de l’espace de variables complexes, Abh. Math. Sern. Univ. Hamburg 11 (1935), pp. 116–162.
17. Krantz S. Harmonic and Complex Analysis in Several Variables, Springer Monographs in Mathematics, Springer, 2017, 1–424.
18. Bremermann H. J. Die Holomorphieh¨ullen der Tuben- und Halbtubengebiete. Math. Ann. 127, 406–423 (1954).
19. Fuks B. A. Special chapters in the theory of analytic functions of several complex variables. Moscow: Nauka, Physical and mathematical literature, 1963. 428 p. (in Russian). Translations of Mathematical Monographs, Vol 14, AMS, 1965, 364 p.
2. Aizenberg L. A., Yuzhakov A. P. Integral Representations and Residues in Multidimensional Complex Analysis, Novosibirsk: Nauka, 335 pp. (1979) (in Russian). Engl. transl.: Transl. Math. Monogr. Vol. 58, Providence, 283 pp. (1983).
3. Khudayberganov G., Kytmanov A. M., Shaimkulov B. A. Analysis in matrix domains. Monograph. Krasnoyarsk, Siberian Federal University, 297 p. (2017) (in Russian).
4. Hua Luogeng. Harmonic analysis of functions of several complex variables in classical domains, Inostr. Lit., M., 1959 (in Russian).
5. Khenkin, G. M. The method of integral representations in complex analysis, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr., 7, VINITI, Moscow, 1985, 23–124 (in Russian).
6. Kytmanov A. M., Nikitina T. N. Analogs of Carleman’s formula for classical domains, Mat. Zametki, 45:3 (1989), 87–93; Math. Notes, 45:3 (1989), 243–248.
7. Khudayberganov G., Abdullayev J. Sh. Relationship between the Kernels Bergman and Cauchy-Szegö in the domains and . J. Sib. Fed. Univ. Math. Phys., 13:5, 559–567 (2020).
8. Khudayberganov G., Rakhmonov U. S. The Bergman and Cauchy–Szegö kernels for matrix ball of the second type, J. Sib. Fed. Univ. Math. Phys., 7:3 (2014), 305–310.
9. Khudayberganov G., Rakhmonov U. S. Carleman Formula for Matrix Ball of the Third Type, Algebra, Complex Analysis, and Pluripotential Theory. USUZCAMP 2017. Springer Proceedings in Mathematics & Statistics, vol. 264, pp. 101–108, Springer, Cham. (2017).
10. Gindikin S. G. Analysis in homogeneous domains, Uspekhi Mat. Nauk, 19:4 (118) (1964), 3–92; Russian Math. Surveys, 19:4 (1964), 1–89.
11. Khudayberganov G., Rakhmonov U. S., Matyakubov Z. Q. Integral formulas for some matrix domains, Topics in Several Complex Variables, AMS, 2016, 89–95.
12. Myslivets S. G. Construction of Szegö and Poisson kernels in convex domains, J. Sib. Fed.Univ. Math. Phys., 2018, Volume 11, Issue 6, 792–795.
13. Rakhmonov U. S., Abdullayev J. Sh. On volumes of matrix ball of third type and generalized Lie balls, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 29:4 (2019), 548–557.
14. Myslivets S. G. On the Szegö and Poisson kernels in the convex domains in , Izv. Vyssh. Uchebn. Zaved. Mat., 2019, no. 1, 42–48.
15. Khudayberganov G., Khalknazarov A. M., Abdullayev J. Sh. Laplace and Hua Luogeng operators, Izv. Vyssh. Uchebn. Zaved. Mat., 2020, no. 3, 74–79.
16. Cartan E. Sur les domaines bornes homogenes de l’espace de variables complexes, Abh. Math. Sern. Univ. Hamburg 11 (1935), pp. 116–162.
17. Krantz S. Harmonic and Complex Analysis in Several Variables, Springer Monographs in Mathematics, Springer, 2017, 1–424.
18. Bremermann H. J. Die Holomorphieh¨ullen der Tuben- und Halbtubengebiete. Math. Ann. 127, 406–423 (1954).
19. Fuks B. A. Special chapters in the theory of analytic functions of several complex variables. Moscow: Nauka, Physical and mathematical literature, 1963. 428 p. (in Russian). Translations of Mathematical Monographs, Vol 14, AMS, 1965, 364 p.
Published
2022-09-26
How to Cite
Shokirovich, J. A., & qizi, K. X. I. (2022). AN ANALOGUE OF BREMERMANN’S THEOREM ON FINDING THE BERGMAN KERNEL FOR THE CARTESIAN PRODUCT OF THE CLASSICAL DOMAINS AND . CENTRAL ASIAN JOURNAL OF MATHEMATICAL THEORY AND COMPUTER SCIENCES, 3(9), 34-43. Retrieved from https://cajmtcs.centralasianstudies.org/index.php/CAJMTCS/article/view/234
Section
Articles
