Wavelets on ultrametric spaces

A family of orthonormal bases, the ultrametric wavelet bases, is introduced in quadratically integrable functions spaces for a wide family of ultrametric spaces. A general family of pseudodifferential operators on this ultrametric spaces is introduced. We show that these operators are diagonal in these ultrametric wavelet bases. A map of considered ultrametric spaces onto real line is discussed. This maps the ultrametric wavelet bases onto orthonormal bases in L2(R+), which are the generalizations of the wavelet bases.