Scaling functions on R2 for dilations of determinant ±2

This paper gives necessary and sufficient conditions for a doubly periodic function p ( ξ ) , ξ ∈ R 2 to be the squared modulus of a lowpass filter for a multiresolution analysis of L 2 ( R 2 ) with respect to an expanding matrix A of determinant ±2. By transferring the underlying spaces, R or R 2 , to a single binary sequence space, we are able to show that, when det ( A ) = 2 , every scaling function on R 2 corresponds to one on R , where the dilation is ±2. If det ( A ) = − 2 , this is no longer true. In this case, the lowpass filter for the stretched Haar function makes an unexpected appearance.