$p$-Adic Haar multiresolution analysis and pseudo-differential operators

Understand

The notion of {\em $p$-adic multiresolution analysis (MRA)} is introduced.

• We discuss 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 a sum of $p$ characteristic functions of mutually disjoint discs of radius $p^{-1}$.
• This refinement equation generates a MRA.