Minimum-energy frames associated with refinable function of arbitrary integer dilation factor

In this paper, we study minimum-energy frame Ψ = {ψ1, ψ2, … , ψN} with arbitrary integer dilation factor d for L2(R), Ψ correspond to some refinable functions with compact support. A precise existence criterion of Ψ is given in terms of an inequality condition on the Laurent polynomial symbols of the refinable functions. We give a constructive proof that when Ψ does exist, d functions with compact support are sufficient to constitute Ψ, and present a explicit formula of constructing Ψ. Finally, we present the minimum-energy frames decomposition and reconstruction formulas which are similar to those of orthogonal wavelets. Numerical examples are given.