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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.

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A class of functional equations and Mielnik probability spaces
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by S. J. Guccione and Č. V. Stanojević PDF
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.
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Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 59 (1976), 317-320
  • MSC: Primary 46C10
  • DOI: https://doi.org/10.1090/S0002-9939-1976-0454605-9
  • MathSciNet review: 0454605