We study p-adic multiresolution analyses (MRAs). A complete characterisation of test functions generating MRAs (scaling functions) is given. We prove that only 1-periodic test functions may be taken as orthogonal scaling functions. We also suggest a method for the construction of wavelet functions and prove that any wavelet function generates a p-adic wavelet frame.

We study the number of Laurent polynomials in a sum of square magnitudesof a nonnegative Laurent polynomial associated with bivariate box splines on the three-and the four-direction meshes. In addition, we study the same problem associated withtrivariate box splines. The number of Laurent polynomials in a sum of squares magnitudeof a nonnegative Laurent polynomial determines the...

We give a characterization of all (quasi) aane frames in L 2 (R n) which have a (quasi) aane dual in terms of the two simple equations in the Fourier transform domain. In particular, if the dual frame is the same as the original system, i.e. it is a tight frame, we obtain the well known characterization of wavelets. Although these equations have already been proven under some special conditions...