We give a characterization of the scaling functions and low pass filters in a translation invariant multiresolution analysis on L2(R). Our conditions involve the notion of locally non-zero function. We write our results in a general context where one considers a dilation given by a fixed expansive linear map on R preserving the integer lattice Z. Indeed, for any such a linear map we construct a scaling function where the support of the Fourier transform is bounded and does not contain any open… Expand