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Pseudodifferential operators on ultrametric spaces and ultrametric wavelets. (English. Russian original) Zbl 1111.35141

Izv. Math. 69, No. 5, 989-1003 (2005); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 69, No. 5, 133-148 (2005).
The authors study a class of pseudodifferential operators on ultrametric spaces. Unfortunately, it is not possible to use a \(p\)-adic Fourier-transform. So an analysis of ultrametric wavelets on ultrametric spaces as the boundary of a directed tree is developed. A family of ultrametrics on directed trees is introduced. The \(p\)-adic distance is generalized onto local compact ultrametric spaces \(X\) over a class of directed trees \(T\). A generalized Haar measure \(\mu\) is introduced. In the corresponding space \(L^2(X,\mu)\) a wavelet basis is constructed. These basic wavelets appear as eigenfunctions of the ultrametric pseudodifferential operator
\[ Tf(x)=\int T(x,y)\bigl(f(x)-f(y)\bigr)d\mu (y) \]
where \(T(x,y)\) is symmetric and has additionally to fullfill local conditions.
It is very nice to see how Kurt Hensel’s number system from 1897 penetrated into many different branches of mathematics.

MSC:

35S05 Pseudodifferential operators as generalizations of partial differential operators
26E30 Non-Archimedean analysis
47G10 Integral operators
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
54E45 Compact (locally compact) metric spaces
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