Necessity of Oversampling Theorem for Affine Frames

Let ,2 an ≥ be two natural numbers. C. K. Chui and X. L. Shi proved that for any affine frame ( ) ( ) { } 2 ;, ,, jj b j k x a a x kb j k Z ψψ = − ∈ of ( ) 2 LR , and the family { } 12 , ,, jk n n j k Z ψ − ∈ is also a frame with the same bounds if n is relatively prime to a . In this paper we prove that n is relatively prime to a which is also necessary.