$p$-Adic Haar multiresolution analysis

Abstract

In this paper, the notion of {\em $p$-adic multiresolution analysis (MRA)} is introduced. We use a ``natural'' refinement equation whose solution (a refinable function) is the characteristic function of the unit disc. This equation reflects the fact that the characteristic function of the unit disc is the sum of $p$ characteristic functions of disjoint discs of radius $p^{-1}$. The case $p=2$ is studied in detail. Our MRA is a 2-adic analog of the real Haar MRA. But in contrast to the real setting, the refinable function generating our Haar MRA is periodic with period 1, which never holds for real refinable functions. This fact implies that there exist infinity many different 2-adic orthonormal wavelet bases in ${\cL}^2(\bQ_2)$ generated by the same Haar MRA. All of these bases are constructed. Since $p$-adic pseudo-differential operators are closely related to wavelet-type bases, our bases can be intensively used for applications.