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A class of functional equations and Mielnik probability spaces

Authors: S. J. Guccione and Č. V. Stanojević
Journal: Proc. Amer. Math. Soc. 59 (1976), 317-320
MSC: Primary 46C10
MathSciNet review: 0454605
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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(\vert\vert x + y\vert\vert),\;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(\vert\vert x + y\vert\vert)$. 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) = \vert\vert x + y\vert\vert{\raise0.5ex\hbox{$\scriptstyle 2$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 4$}}$, we answer a question due to B. Mielnik.

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Keywords: Mielnik probability spaces, functional equation
Article copyright: © Copyright 1976 American Mathematical Society

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