| Title:
|
The Bergman kernel: Explicit formulas, deflation, Lu Qi-Keng problem and Jacobi polynomials (English) |
| Author:
|
Beberok, Tomasz |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
67 |
| Issue:
|
2 |
| Year:
|
2017 |
| Pages:
|
537-549 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We investigate the Bergman kernel function for the intersection of two complex ellipsoids $\{(z,w_1,w_2) \in \mathbb {C}^{n+2} \colon |z_1|^2 + \cdots + |z_n|^2 + |w_1|^q < 1, \ |z_1|^2 + \cdots + |z_n|^2 + |w_2|^r < 1\}. $ We also compute the kernel function for $\{(z_1,w_1,w_2) \in \mathbb {C}^3 \colon |z_1|^{2/n} + |w_1|^q < 1, \ |z_1|^{2/n} + |w_2|^r < 1\}$ and show deflation type identity between these two domains. Moreover in the case that $q=r=2$ we express the Bergman kernel in terms of the Jacobi polynomials. The explicit formulas of the Bergman kernel function for these domains enables us to investigate whether the Bergman kernel has zeros or not. This kind of problem is called a Lu Qi-Keng problem. (English) |
| Keyword:
|
Lu Qi-Keng problem |
| Keyword:
|
Bergman kernel |
| Keyword:
|
Routh-Hurwitz theorem |
| Keyword:
|
Jacobi polynomial |
| MSC:
|
32A25 |
| MSC:
|
33D70 |
| idZBL:
|
Zbl 06738537 |
| idMR:
|
MR3661059 |
| DOI:
|
10.21136/CMJ.2017.0073-16 |
| . |
| Date available:
|
2017-06-01T14:32:45Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146774 |
| . |
| Reference:
|
[1] Beberok, T.: Lu Qi-Keng’s problem for intersection of two complex ellipsoids.Complex Anal. Oper. Theory 10 (2016), 943-951. Zbl 06601600, MR 3506300, 10.1007/s11785-015-0505-4 |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
[5] Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F. G.: Higher Transcendental Functions.Vol. 1, Bateman Manuscript Project, McGraw-Hill, New York (1953). Zbl 0051.30303, MR 0698779 |
| Reference:
|
[6] Krantz, S. G.: Geometric Analysis of the Bergman Kernel and Metric.Graduate Texts in Mathematics 268, Springer, New York (2013). Zbl 1281.32004, MR 3114665, 10.1007/978-1-4614-7924-6 |
| Reference:
|
[7] Šiljak, D. D., Stipanović, D. M.: Stability of interval two-variable polynomials and quasipolynomials via positivity.Positive Polynomials in Control D. Henrion, et al. Lecture Notes in Control and Inform. Sci. 312, Springer, Berlin (2005), 165-177. Zbl 1138.93392, MR 2123523, 10.1007/10997703_10 |
| Reference:
|
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