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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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