Geodesic
Regularization of a~three-element functional equation
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 4 (2009), pp. 39-42

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In this paper we study the three-element functional equation $$ (V\Phi)(z)\equiv\Phi(iz)+\Phi(-iz)+G(z)\Phi\biggl(\frac1z\biggr)=g(z),\qquad z\in R, $$ 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. 
@article{IVM_2009_4_a4,
     author = {S. A. Modina},
     title = {Regularization of a~three-element functional equation},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {39--42},
     publisher = {mathdoc},
     number = {4},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2009_4_a4/}
}
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S. A. Modina. Regularization of a~three-element functional equation. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 4 (2009), pp. 39-42. http://geodesic.mathdoc.fr/item/IVM_2009_4_a4/