# Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

## A class of functional equations and Mielnik probability spacesHTML articles powered by AMS MathViewer

by S. J. Guccione and Č. V. Stanojević
Proc. Amer. Math. Soc. 59 (1976), 317-320 Request permission

## 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 $p(x,y) = f(||x + y||),\;x,\;y \in S$, where $f$ 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 $\mathcal {B}$, with $p = \phi \circ f$ where $\phi$ is a solution of a certain functional equation. In particular, for $p(x,y) = ||x + y||^2/4$, we answer a question due to B. Mielnik.
References
Similar Articles
• Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46C10
• Retrieve articles in all journals with MSC: 46C10