The Bergman kernel: Explicit formulas, deflation, Lu Qi-Keng problem and Jacobi polynomials

Tomasz Beberok

Czechoslovak Mathematical Journal (2017)

  • Volume: 67, Issue: 2, page 537-549
  • ISSN: 0011-4642

Abstract

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We investigate the Bergman kernel function for the intersection of two complex ellipsoids { ( z , w 1 , w 2 ) ∈ ℂ n + 2 : | z 1 | 2 + ⋯ + | z n | 2 + | w 1 | q < 1 , | z 1 | 2 + ⋯ + | z n | 2 + | w 2 | r < 1 } . We also compute the kernel function for { ( z 1 , w 1 , w 2 ) ∈ ℂ 3 : | 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.

How to cite

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Beberok, Tomasz. "The Bergman kernel: Explicit formulas, deflation, Lu Qi-Keng problem and Jacobi polynomials." Czechoslovak Mathematical Journal 67.2 (2017): 537-549. <http://eudml.org/doc/288187>.

@article{Beberok2017,
abstract = {We investigate the Bergman kernel function for the intersection of two complex ellipsoids $\lbrace (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\rbrace . $ We also compute the kernel function for $\lbrace (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\rbrace $ 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.},
author = {Beberok, Tomasz},
journal = {Czechoslovak Mathematical Journal},
keywords = {Lu Qi-Keng problem; Bergman kernel; Routh-Hurwitz theorem; Jacobi polynomial},
language = {eng},
number = {2},
pages = {537-549},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The Bergman kernel: Explicit formulas, deflation, Lu Qi-Keng problem and Jacobi polynomials},
url = {http://eudml.org/doc/288187},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Beberok, Tomasz
TI - The Bergman kernel: Explicit formulas, deflation, Lu Qi-Keng problem and Jacobi polynomials
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 2
SP - 537
EP - 549
AB - We investigate the Bergman kernel function for the intersection of two complex ellipsoids $\lbrace (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\rbrace . $ We also compute the kernel function for $\lbrace (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\rbrace $ 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.
LA - eng
KW - Lu Qi-Keng problem; Bergman kernel; Routh-Hurwitz theorem; Jacobi polynomial
UR - http://eudml.org/doc/288187
ER -

References

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  6. Krantz, S. G., 10.1007/978-1-4614-7924-6, Graduate Texts in Mathematics 268, Springer, New York (2013). (2013) Zbl1281.32004MR3114665DOI10.1007/978-1-4614-7924-6
  7. Šiljak, D. D., Stipanović, D. M., 10.1007/10997703_10, Positive Polynomials in Control D. Henrion, et al. Lecture Notes in Control and Inform. Sci. 312, Springer, Berlin (2005), 165-177. (2005) Zbl1138.93392MR2123523DOI10.1007/10997703_10
  8. Wang, X., 10.1080/10652469.2011.596483, Integral Transforms Spec. Funct. 23 (2012), 421-433. (2012) Zbl1273.33013MR2929185DOI10.1080/10652469.2011.596483

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