Geodesic
Wavelet theory as $p$-adic spectral analysis
Izvestiya. Mathematics , Tome 66 (2002) no. 2, pp. 367-376

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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. 
@article{IM2_2002_66_2_a3,
     author = {S. V. Kozyrev},
     title = {Wavelet theory as $p$-adic spectral analysis},
     journal = {Izvestiya. Mathematics },
     pages = {367--376},
     publisher = {mathdoc},
     volume = {66},
     number = {2},
     year = {2002},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2002_66_2_a3/}
}
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S. V. Kozyrev. Wavelet theory as $p$-adic spectral analysis. Izvestiya. Mathematics , Tome 66 (2002) no. 2, pp. 367-376. http://geodesic.mathdoc.fr/item/IM2_2002_66_2_a3/