On orthogonal p-adic wavelet bases

A variety of different orthogonal wavelet bases has been found for L2(R)L2(R) for the last three decades. Such bases are not only interesting by themselves, a number of theoretical and applied problems were solved using them. It appeared that similar non-trivial wavelet constructions also exist for functions defined on some other algebraic structures, such as local fields of positive characteristic including the Vilenkin/Cantor groups. We note that most of the bases involved consist of Bruhat–Schwartz functions, i.e. compactly supported and band-limited ones. There were several attempts to develop wavelet theory in a more general setting, e.g., for zero-dimensional groups. Analyzing these papers, one can see that actually nothing except for the Haar basis and its trivial modifications was constructed. In the present paper we give an explanation for these failures. Namely, we describe the situation for the field QpQp of p  -adic numbers and prove that any orthogonal wavelet basis for L2(Qp)L2(Qp) that consists of band-limited (periodic) functions, is a modification of the Haar basis.