MÖbius Transformations and Multiplicative Representations for Spherical Potentials
Publications de l'Institut Mathématique, _N_S_75 (2004) no. 89, p. 253
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For the unit spheres $S^n\subset\mathbf R^{n+1}$ and
$S^{2n-1}\subset\mathbf R^{2n}=\mathbf C^n$ we prove the following
identities for two classical potentials
$
\int_{S^n}\frac{f(y)}{|x-y|^{n+\alpha}}d\sigma_y =\frac{1}{|1-|x|^2|^\alpha}
\int_{S^n}\frac{f(T_{n,x}(y))}{|x-y|^{n-\alpha}}d\sigma_y,
$
$
\int_{S^{2n-1}}\frac{F(\zeta)d\sigma_\zeta}{|1-(z,\zeta)|^{n+\alpha}}=
\frac{1}{(1-|z|^2)^\alpha}\int_{S^{2n-1}}
\frac{F(\Phi_{n,z}(\zeta))d\sigma_\zeta}{|1-(z,\zeta)|^{n-\alpha}},
$
where $x\in\mathbf R^{n+1}$ ($|x|\ne0$ and $|x|\ne1$),
$z\in\mathbf C^n$ ($|z|1$), $T_{n,x}$ and $\Phi_{n,z}$ are explicit
involutions of $S^n$ and $S^{2n-1}$ respectively. Some applications of
these formulas are also considered.
F. G. Avkhadiev. MÖbius Transformations and Multiplicative Representations for Spherical Potentials. Publications de l'Institut Mathématique, _N_S_75 (2004) no. 89, p. 253 . doi: 10.2298/PIM0475253A
@article{10_2298_PIM0475253A,
author = {F. G. Avkhadiev},
title = {M\"Obius {Transformations} and {Multiplicative} {Representations} for {Spherical} {Potentials}},
journal = {Publications de l'Institut Math\'ematique},
pages = {253 },
year = {2004},
volume = {_N_S_75},
number = {89},
doi = {10.2298/PIM0475253A},
zbl = {1078.31004},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.2298/PIM0475253A/}
}
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%0 Journal Article %A F. G. Avkhadiev %T MÖbius Transformations and Multiplicative Representations for Spherical Potentials %J Publications de l'Institut Mathématique %D 2004 %P 253 %V _N_S_75 %N 89 %U http://geodesic.mathdoc.fr/articles/10.2298/PIM0475253A/ %R 10.2298/PIM0475253A %G en %F 10_2298_PIM0475253A
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