Wavelet theory as $p$-adic spectral analysis
Abstract: We construct a new orthonormal basis of eigenfunctions of the Vladimirov $p$-adic fractional differentiation operator. We construct a map of the $p$-adic numbers onto the real numbers (the $p$-adic change of variables), which transforms the Haar measure on the $p$-adic numbers to the Lebesgue measure on the positive semi-axis. The $p$-adic change of variables (for $p=2$) provides an equivalence between the basis of eigenfunctions of the Vladimirov operator and the wavelet basis in $L^2({\mathbb R}_+)$ generated by the Haar wavelet. This means that wavelet theory can be regarded as $p$-adic spectral analysis.
Received: 23.02.2001
Document Type: Article
UDC: 517.58+517.53.02
Language: English
Original paper language: Russian
Citation: S. V. Kozyrev, “Wavelet theory as $p$-adic spectral analysis”, Izv. Math., 66:2 (2002), 367–376
Citation in format AMSBIB\Bibitem{Koz02}
\by S.~V.~Kozyrev
\paper Wavelet theory as $p$-adic spectral analysis
\jour Izv. Math.
\yr 2002
\vol 66
\issue 2
\pages 367--376
\mathnet{http://mi.mathnet.ru/eng/im381}
\crossref{https://doi.org/10.1070/IM2002v066n02ABEH000381}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1918846}
\zmath{https://zbmath.org/?q=an:1016.42025}
\elib{https://elibrary.ru/item.asp?id=14114380}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-3843060681}
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