Abstract
Let S be the unit sphere of a normed real linear space N and let (S, p) be a Mielnik space of dimension two. For p(x, y) = f(‖x+y‖), x, yєS, where /is a continuous, strictly increasing function from [0, 2] onto [0, 1], it has been shown that (S, p) being two dimensional is equivalent to N being an inner product space. In some polarization problems modeled on the unit sphere of an inner product space, the transition probability p(x, y) may not be as well behaved as p(x, y) = f(‖x + y‖). In order to provide a more suitable setting, we have constructed wide classes of two-dimensional transitional probability spaces (S, p), all having the same set of bases ℬ with p=⌽ ° f where ⌽ is a solution of a certain functional equation. In particular, for p(x, y) = ‖x+y‖24, we answer a question due to B. Mielnik. © American Mathematical Society 1976.
Recommended Citation
S. J. Guccione and C. V. Stanojevic, "A Class Of Functional Equations And Mielnik Probability Spaces," Proceedings of the American Mathematical Society, vol. 59, no. 2, pp. 317 - 320, American Mathematical Society, Jan 1976.
The definitive version is available at https://doi.org/10.1090/S0002-9939-1976-0454605-9
Department(s)
Mathematics and Statistics
Keywords and Phrases
Functional equation; Mielnik probability spaces
International Standard Serial Number (ISSN)
1088-6826; 0002-9939
Document Type
Article - Journal
Document Version
Citation
File Type
text
Language(s)
English
Rights
© 2023 American Mathematical Society, All rights reserved.
Publication Date
01 Jan 1976
