L2(R) solutions of dilation equations and Fourier-like transforms

We state a novel construction of theFourier transform on L2(R) based on translation and dilation properties which makes use of the multiresolution analysis structure commonly used in the design of wavelets. We examine the conditions imposed by variants of these translation and dilation properties. This allows other characterizations of the Fourier transform to be given, and operators which have similar properties are classified. This is achieved by examining the solution space of various dilation equations, in particular we show that the L2(R) solutions of f (x) = f (2x) + f (2x − 1) are in direct correspondence with L2(±[1, 2))