From scaling sets to scaling functions

The paper presents a general method for construction of scaling functions in Rn for an arbitrary expanding matrix with integer coefficients. Using a scaling set as a starting point, values of the corresponding characteristic function are modified in a way that obtained object still remains the Fourier transform of a scaling function. Moreover, it is shown that every MRA wavelet can be constructed using this procedure. Several examples are included. In particular, examples of non-MSF orthonormal wavelets for any integer dilation factor on the real line and for the quincunx matrix in R2 are demonstrated.