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 Izv. Vyssh. Uchebn. Zaved. Mat., 2009, Number 4, Pages 39–42 (Mi ivm1318)

Regularization of a three-element functional equation

S. A. Modina

Kazan State Power Engineering University

Abstract: In this paper we study the three-element functional equation
$$(V\Phi)(z)\equiv\Phi(iz)+\Phi(-iz)+G(z)\Phi(\frac1z)=g(z),\qquad z\in R,$$
subject to
$$R\colon |z|<1,\quad|\arg z|<\frac\pi4.$$

We assume that the coefficients $G(z)$ and $g(z)$ are holomorphic in $R$ and their boundary values $G^+(t)$ and $g^+(t)$ belong to $H(\Gamma)$, $G(t)G(t^{-1})=1$. We seek for solutions $\Phi(z)$ in the class of functions holomorphic outside of $\overline R$ such that they vanish at infinity and their boundary values $\Phi^-(t)$ also belong to $H(\Gamma)$.
Using the method of equivalent regularization, we reduce the problem to the 2nd kind integral Fredholm equation.

Keywords: functional equation, holomorphic function, regularization method, rotation group of a dihedron.

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English version:
Russian Mathematics (Izvestiya VUZ. Matematika), 2009, 53:4, 31–33

Bibliographic databases:

UDC: 517.51

Citation: S. A. Modina, “Regularization of a three-element functional equation”, Izv. Vyssh. Uchebn. Zaved. Mat., 2009, no. 4, 39–42; Russian Math. (Iz. VUZ), 53:4 (2009), 31–33

Citation in format AMSBIB
\Bibitem{Mod09} \by S.~A.~Modina \paper Regularization of a~three-element functional equation \jour Izv. Vyssh. Uchebn. Zaved. Mat. \yr 2009 \issue 4 \pages 39--42 \mathnet{http://mi.mathnet.ru/ivm1318} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2581472} \transl \jour Russian Math. (Iz. VUZ) \yr 2009 \vol 53 \issue 4 \pages 31--33 \crossref{https://doi.org/10.3103/S1066369X09040057}