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Trigonometric convexity of the multidimensional indicator

Part of: Harmonic analysis in several variables Analytic spaces Holomorphic functions of several complex variables

Published online by Cambridge University Press:  05 January 2024

Aleksandr Mkrtchyan  and
Armen Vagharshakyan Open the ORCID record for Armen Vagharshakyan [Opens in a new window]
Aleksandr Mkrtchyan
Affiliation:
Siberian Federal University, Krasnoyarsk, Russia, Institute of Mathematics NAS, Yerevan, Armenia and Saint Petersburg University, Saint Petersburg, Russia e-mail: AMkrtchyan@sfu-kras.ru
Armen Vagharshakyan*
Affiliation:
Institute of Mathematics NAS, Yerevan, Armenia and Yerevan State University, Yerevan, Armenia
*
e-mail: avaghars@kent.edu
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Abstract

The notion of indicator of an analytic function, that describes the function’s growth along rays, was introduced by Phragmen and Lindelöf. Trigonometric convexity is a defining property of the indicator. For multivariate cases, an analogous property of trigonometric convexity was not known so far. We prove the property of trigonometric convexity for the indicator of multivariate analytic functions, introduced by Ivanov. The results that we obtain are sharp. Derivation of a multidimensional analogue of the inverse Fourier transform in a sector and obtaining estimates on its decay is an important step of our proof.

Keywords

Analytic functions in several complex variablesFourier transformindicator functionanalytic continuationtrigonometric convexity

MSC classification

Primary: 32A26: Integral representations, constructed kernels (e.g. Cauchy, Fantappiè-type kernels)
Secondary: 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 32C30: Integration on analytic sets and spaces, currents
Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 16
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

The first author was funded by the Saint Petersburg Leonhard Euler International Mathematical Institute and supported by the Ministry of Science and Higher Education of the Russian Federation (Agreement No. 075-15-2022-287).

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