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 Vladikavkaz. Mat. Zh., 2017, Volume 19, Number 1, Pages 3–10 (Mi vmj601)

A reconstruction of analytic functions on the unit disk of $\mathbb{C}$

I. I. Bavrina, O. E. Yaremkob

a Moscow State Pedagogical University
b Penza State Pedagogical University named after V. G. Belinsky

Abstract: A reconstruction of an analytic function on the unit disk of $\mathbb{C}$ by its integral characteristics is studied. An algorithm for solving the inverse problem of integral geometry in the space of analytic functions in the unit disk is presented. The main idea behind the paper is that the reconstruction method is determined by the class of functions under consideration: The narrower the class of functions is, the less information one needs to know to restore the function. The simplest reconstruction formulas are obtained in the class of analytic functions in the unit disk. Three integral representations for analytic functions in the unit disk are established. The first formula reconstructs the function by its means along radii. The second one restores an analytic function in the unit ball by its means along the vertical line segments. The third integral formula reconstructs an analytic function from its weighted means on the circle.

Key words: the Radon transform, integral representation, the inverse problem, the Legendre polynomial.

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Citation: I. I. Bavrin, O. E. Yaremko, “A reconstruction of analytic functions on the unit disk of $\mathbb{C}$”, Vladikavkaz. Mat. Zh., 19:1 (2017), 3–10

Citation in format AMSBIB
\Bibitem{BavYar17} \by I.~I.~Bavrin, O.~E.~Yaremko \paper A reconstruction of analytic functions on the unit disk of $\mathbb{C}$ \jour Vladikavkaz. Mat. Zh. \yr 2017 \vol 19 \issue 1 \pages 3--10 \mathnet{http://mi.mathnet.ru/vmj601} 

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