Discrete autocorrelation-based multiplicative MRAs and sampling on R

GivenK > 0 and a positive integerM, the function△K,M (t) = Kmax(1− |t| M , 0) represents an isosceles triangle on the real line R having height K and base 2M. For a given K and M, a constant amplitude sequence is constructed on the integers Z having autocorrelation △K,M restricted to Z. With this setup, a large class of bounded sequences z is shown to exist, where the autocorrelation of z is a finite sum of such △K,Ms. By taking an inverse Fourier transform we are led naturally to expansions in terms of Fejér functions. This line of thought is closely related to the construction of constant amplitude zero autocorrelation (CAZAC) sequences which have applications in radar and communications. Extending the setting to R, a Fejér function decomposition formula is obtained for a class of even continuous functions. This gives rise to a decomposition theorem along the lines of what can be described as multiplicative multiresolution analysis (MRA) and sampling theory. The underlying techniques generalize to other generating functions besides the Fejér function, even for the case of R.