Functional Equation with Holomorphic Coefficients Related to a Finite Group

Let D be a semicircle, which is the fundamental domain of a finite properly discontinuous group of linear-fractional transformations. We consider a linear four-element functional equation with holomorphic coefficients in D that is generated by this group. The solution is sought in the class of funct...

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Published in:Russian mathematics 2022-08, Vol.66 (8), p.27-30
Main Authors: Garif’yanov, F. N., Strezhneva, E. V.
Format: Article
Language:English
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Entire functions
Fixed points (mathematics)
Functional equations
Group theory
Mathematical analysis
Mathematics
Mathematics and Statistics
Regularization
Transformations (mathematics)
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Strezhneva, E. V.
description Let D be a semicircle, which is the fundamental domain of a finite properly discontinuous group of linear-fractional transformations. We consider a linear four-element functional equation with holomorphic coefficients in D that is generated by this group. The solution is sought in the class of functions that are holomorphic outside “half” the boundary ∂ D and vanishing at infinity. To regularize the equation, we introduce a Carleman involutive shift induced by the generating transformations of the group. This shift has two fixed points. The condition for regularization equivalence is found. Applications to the problem of moments for entire functions of exponential type are specified.
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subjects Entire functions
Fixed points (mathematics)
Functional equations
Group theory
Mathematical analysis
Mathematics
Mathematics and Statistics
Regularization
Transformations (mathematics)
title Functional Equation with Holomorphic Coefficients Related to a Finite Group
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