Orthogonal Gyroexpansion in Möbius Gyrovector Spaces

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.

A Cauchy type inequality for Möbius operations

“…The following theorem is a criterion for convergence of series in Möbius addition, which subsumes [ 6 , Example 34].…”

“…Then by the definition of the function (see [ 2 , (6.288), p. 216]), the definition of the partial sum in the Möbius addition, commutativity and associativity of ⊕ on the interval , , and the assumption . Since the space is complete (see [ 6 , Lemma 20 and Theorem 26]), there exists a value such that , which implies that . This completes the proof.…”

On Quasi Gyrolinear Maps between Möbius Gyrovector Spaces Induced from Finite Matrices

“…Lemma 1. [15] (Lemma 31). If {u, v, w} is an orthogonal set in V s , then the associative law holds; that is,…”

“…The following identities are an easy consequence of the definition, and frequently used. One can refer to [15] (Lemma 12, Lemma 14 (i)).…”

“…In recent years, we have clarified the structure of Möbius gyrovector spaces to some extent, such as the structure of finitely generated gyrovector subspaces, orthogonal gyrodecomposition of any element with respect to any closed gyrovector subspace, orthogonal gyroexpansion of any element with respect to an arbitrary orthogonal basis with weight sequence, Cauchy-Schwarz-type inequalities, and continuous quasi gyrolinear functionals induced by any square summable sequence of real numbers (cf. [14][15][16][17][18][19]). The purpose of this article is to present a class of continuous maps between Möbius gyrovector spaces induced by finite matrices, which can be regarded as a certain counterpart to bounded linear operators on real Hilbert spaces.…”