Wavelet theory as $p$-adic spectral analysis

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.