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Unboundedness of a \(p\)-adic Gaussian distribution. (English. Russian original) Zbl 0819.46063
Russ. Acad. Sci., Izv., Math. 41, No. 2, 367-375 (1993); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 56, No. 5, 1104-1115 (1992).
Let \(\mathbb{Z}_ p\) be the ring of \(p\)-adic integers. The assignment \[ \lambda:\chi^ n\mapsto\begin{cases} {(2k)!\over k!}({b\over 2})^ k\quad &\text{if }n= 2k,\;k\in\mathbb{N}\\ 0\quad &\text{if }n\text{ is odd}\end{cases} \] where \(b\) is a \(p\)-adic number, defines a linear function on the ring of polynomial functions on \(\mathbb{Z}_ p\) (with \(p\)-adic values), called a Gaussian distribution. (Such distributions appeared in previous work of the author on \(p\)-adic quantum mechanics.) If \({b\over 2}\in \mathbb{Z}_ p\) \((b\neq 0)\) then \(\lambda\) can in a natural way be extended to the space of all analytic functions on \(\mathbb{Z}_ p\).
Is this also the case for the space of continuous functions?
The authors show that, at least for \(p\neq 2\), the answer is ‘no’ by a rather non-trivial and involved proof of the fact that the sequence \(n\mapsto\lambda{\chi\choose n}\) is unbounded \(p\)-adically.

MSC:
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
11K41 Continuous, \(p\)-adic and abstract analogues
81Q99 General mathematical topics and methods in quantum theory
14G20 Local ground fields in algebraic geometry
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