A general framework for extending means to higher orders
By Jimmie Lawson, Yongdo Lim. 2006
55 citations recorded by the source corpus.
Although there is an extensive literature on various means of two positive operators and their applications, these means do not typically readily extend to means of three and more operators. It has been an open problem to dene and prove the existence of these higher order means in a general setting. In this paper we lay the foundations for such a theory by showing how higher order means can be inductively dened and established in general metric spaces, in particular, in convex metric spaces. We consider uniqueness properties and preservation properties of these extensions, properties which provide validation to our approach. As our targeted application, we consider the positive operators on a Hilbert space under the Thompson metric and apply our methods to derive higher order extensions of a variety of standard operator means such as the geometric mean, the Gauss mean, and the logarithmic mean. That the operator logarithmic mean admits extensions of all higher orders provides a positive solution to a problem of Petz and Temesi (SIAM J. Matrix Anal. Appl. 27 (2005)).
Subfields: Applied Mathematics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/f33d1cb476a07d002433fffecd180a828dbd4c0d
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In Applied Mathematics
In Statistics and Probability
Citations in the corpus
- Monotonic properties of the least squares mean (2010, methodology): “Under appropriate metric hypotheses, all of which are implied by the NPC condition, the symmetrization procedure applies and inductively yields multivariable means extending x#y for each n ≥ 3; see Es-Sahib and Heinrich [11] and the authors [20].”