The Construction of Minimum-Energy Frames with Arbitrary integer Dilation Factor

In this paper, we study minimum-energy frame Ψ = {ψ¹, ψ², Λ, ψ^N} with compact support for L²(R), and with arbitrary integer dilation factor d, Ψ correspond to some compactly supported refinable functions, ,give a precise existence criterion of Ψ in terms of an inequality condition on the Laurent polynomial symbols of the refinable function. When Ψ does exist, d compactly supported functions are sufficient to constitute Ψ, and present a explicit formula of constructing Ψ. Numerical examples are given.