Orthogonal Gyroexpansion in Möbius Gyrovector Spaces

We investigate the Mobius gyrovector spaces which are open balls centered at the origin in a real Hilbert space with the Mobius addition, the Mobius scalar multiplication, and the Poincare 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 Mobius 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.