Supports of Fourier transforms of scaling functions

We characterize the support G of the Fourier transform of the band-limited scaling function and give an approach to the construction of the scaling function. Based on the relations of translates of three point sets G, and , we reveal an essential difference of scaling functions corresponding to MRA, weakly translation invariant MRA and translation invariant MRA. Moreover, we consider the case that support of the Fourier transform of the scaling function is a convex set. Finally, we show that the support of Fourier transform of any scaling function cannot be a closed ball.