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Non-Archimedean analysis and a wave-type pseudodifferential equation on finite adèles
- Source :
- Journal of Pseudo-Differential Operators and Applications. 11:1139-1181
- Publication Year :
- 2020
- Publisher :
- Springer Science and Business Media LLC, 2020.
-
Abstract
- In this work the ring of finite adeles $${\mathbb {A}}_f$$ of the rational numbers $${\mathbb {Q}}$$ is obtained as a completion of $${\mathbb {Q}}$$ with respect to a certain non-Archimedean metric related to the second Chebyshev function, which allows us to represent any finite adele as a convergent series, generalizing m-adic analysis. This polyadic analysis allows us to introduce a novel pseudodifferential operator $$D^{\alpha }$$ on $$L^2({\mathbb {A}}_f)$$ of fractional differentiation, similar to the Vladimirov operator on the p-adic numbers. The operator $$D^{\alpha }$$ is a positive selfadjoint unbounded operator whose spectrum $$\sigma (D^{\alpha })$$ is essential and it consists of a countable number of eigenvalues, which converges to zero, and zero. Moreover, a sort of multiresolution analysis on $${\mathbb {A}}_f$$ provides us with a wavelet basis which is an orthonormal basis of eigenfunctions of $$D^{\alpha }$$ as well. The Cauchy problem of a wave-type pseudodifferential equation $$\begin{aligned} u_{tt}(x,t)+D^{\alpha }_x u(x,t) =F(x,t), \qquad (x \in {\mathbb {A}}_f), \end{aligned}$$ with appropriate initial conditions $$u(x,0)=f(x), u_t(x,0)=g(x),$$ and external force F(x, t), is solved separating variables and using the Fourier expansion of functions in $$L^2({\mathbb {A}}_f)$$ , with respect to the wavelet basis.
- Subjects :
- Unbounded operator
Rational number
Applied Mathematics
010102 general mathematics
Spectrum (functional analysis)
Zero (complex analysis)
Type (model theory)
Eigenfunction
Operator theory
01 natural sciences
010101 applied mathematics
Combinatorics
0101 mathematics
Analysis
Convergent series
Mathematics
Subjects
Details
- ISSN :
- 1662999X and 16629981
- Volume :
- 11
- Database :
- OpenAIRE
- Journal :
- Journal of Pseudo-Differential Operators and Applications
- Accession number :
- edsair.doi...........608022f2353bcc7ab84338582dd1fd30