Abstract
Let \( \mathbb{D} \) be an open unit disk in the complex plane. It is shown that every subspace in C(\( \mathbb{D} \)) invariant under weighted conformal shifts contains a radial eigenfunction of the corresponding invariant differential operator. This function can be expressed via the Gauss hypergeometric function and is a generalization of the spherical function on the disk \( \mathbb{D} \) which is considered as a hyperbolic plane with the corresponding Riemannian structure.
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Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 12, No. 3, pp. 326–344, July–August, 2015.
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Volchkov, V.V., Volchkov, V.V. An analog of the Schwartz theorem on spectral analysis on a hyperbolic plane. J Math Sci 214, 172–185 (2016). https://doi.org/10.1007/s10958-016-2767-4
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DOI: https://doi.org/10.1007/s10958-016-2767-4