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Journal of Function Spaces
Volume 2017, Article ID 1518254, 13 pages
https://doi.org/10.1155/2017/1518254
Research Article

Orthogonal Gyroexpansion in Möbius Gyrovector Spaces

Department of Mathematics, Faculty of Science, Niigata University, Niigata 950–2181, Japan

Correspondence should be addressed to Keiichi Watanabe; pj.ca.u-atagiin.cs.htam@kbntw

Received 12 August 2017; Accepted 17 September 2017; Published 13 December 2017

Academic Editor: Juan Martinez-Moreno

Copyright © 2017 Keiichi Watanabe. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate the Möbius gyrovector spaces which are open balls centered at the origin in a real Hilbert space with the Möbius addition, the Möbius scalar multiplication, and the Poincaré metric introduced by Ungar. In particular, for an arbitrary point, we can easily obtain the unique closest point in any closed gyrovector subspace, by using the ordinary orthogonal decomposition. Further, we show that each element has the orthogonal gyroexpansion with respect to any orthogonal basis in a Möbius gyrovector space, which is similar to each element in a Hilbert space having the orthogonal expansion with respect to any orthonormal basis. Moreover, we present a concrete procedure to calculate the gyrocoefficients of the orthogonal gyroexpansion.