A class of functional equations and Mielnik probability spaces

Guccione, S. J.; Stanojević, Č. V.

doi:10.1090/S0002-9939-1976-0454605-9

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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