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Determining sets for holomorphic functions on the symmetrized bidisk

Part of: Holomorphic functions of several complex variables Linear function spaces and their duals Holomorphic convexity Analytic spaces

Published online by Cambridge University Press:  31 January 2023

Bata Krishna Das ,
Poornendu Kumar  and
Haripada Sau
Bata Krishna Das
Affiliation:
Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India e-mail: dasb@math.iitb.ac.in bata436@gmail.com
Poornendu Kumar
Affiliation:
Department of Mathematics, Indian Institute of Science, Bengaluru 560012, India e-mail: poornendukumar@gmail.com poornenduk@iisc.ac.in
Haripada Sau*
Affiliation:
Department of Mathematics, Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pashan, Pune, Maharashtra 411008, India
*
e-mail: haripadasau215@gmail.com
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Abstract

A subset ${\mathcal D}$ of a domain $\Omega \subset {\mathbb C}^d$ is determining for an analytic function $f:\Omega \to \overline {{\mathbb D}}$ if whenever an analytic function $g:\Omega \rightarrow \overline {{\mathbb D}}$ coincides with f on ${\mathcal D}$, equals to f on whole $\Omega $. This note finds several sufficient conditions for a subset of the symmetrized bidisk to be determining. For any $N\geq 1$, a set consisting of $N^2-N+1$ many points is constructed which is determining for any rational inner function with a degree constraint. We also investigate when the intersection of the symmetrized bidisk intersected with some special algebraic varieties can be determining for rational inner functions.

Keywords

Rational inner functionssymmetrized bidiskdistinguished varietiesNevanlinna–Pick interpolationuniqueness varietiesdetermining setsextension problemreproducing kernel Hilbert spaces

MSC classification

Primary: 32A10: Holomorphic functions
Secondary: 32A70: Functional analysis techniques 32C25: Analytic subsets and submanifolds 32E30: Holomorphic and polynomial approximation, Runge pairs, interpolation 46E22: Hilbert spaces with reproducing kernels (=
Type
Article
Information
Canadian Mathematical Bulletin , Volume 66 , Issue 3 , September 2023 , pp. 984 - 996
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

B.K.D. is supported by the Mathematical Research Impact Centric Support (MATRICS) grant, File No: MTR/2021/000560, by the Science and Engineering Research Board (SERB), Department of Science & Technology (DST), Government of India. P.K. is supported by the University Grants Commission Centre for Advanced Studies. The research works of H.S. is supported by DST-INSPIRE Faculty Fellowship DST/INSPIRE/04/2018/002458.

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