Coincidence of the continuous and discrete p – adic wavelet transforms

We show that translations and dilations of a p–adic wavelet coincides (up to the multiplication by some root of one) with a vector from the known basis of discrete p–adic wavelets. In this sense the continuous p–adic wavelet transform coincides with the discrete p–adic wavelet transform. The p–adic multiresolution approximation is introduced and relation with the real multiresolution approximation is described. Relation of application of p–adic wavelets to spectral theory of p–adic pseudodiffer-ential operators and the known results about sparsity of matrices of some real integral operators in the bases of multiresolution wavelets is discussed.