Abstract
We present a basis of p-adic wavelets for Sobolev-type spaces consisting of eigenvectors of certain pseudodifferential operators. Our result extends a well-known result due to S. Kozyrev.
Highlights
The field of p-adic numbers was introduced by the German mathematician Kurt Hensel in 1897
In this article we present a basis of p−adic wavelets for Sobolev-type spaces Hl (C) with l ∈ N, see Theorem 3.6
By using (12), we obtain that the product of indicators is non-zero: Ω( p−γ ξ + p−1ζ p)Ω( ξ p) = Ω( p−γ ξ + p−1ζ p)Ω( − pγ−1ζ p), if γ ≥ 1
Summary
The field of p-adic numbers was introduced by the German mathematician Kurt Hensel in 1897. Volovich posed the conjecture of the non-Archimedean nature of the space-time at the level of the Planck scale. This conjecture has originated a lot of research, for instance, in quantum mechanics, see e.g. The spaces Hl(C) were introduced in [16], these spaces are the completion of the C-vector space of Bruhat-Schwartz functions with respect to an inner product ·, · l , l ∈ N, (which coincides with the product of L2 when l = 0) These spaces are very important in the construction of the non-. Archimedean versions of the Kondratiev and Hida spaces, which in turn are useful in the construction of quantum field theories over a p−adic space-time, see [3]. We show that the functions ψγ(l,)η,ζ, are eigenfunctions for a pseudodifferential operator with a radial symbol
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