𝑛× oversampling preserves any tight affine frame for odd 𝑛

Abstract

If ψ \psi generates an affine frame ψ j , k ( x ) = 2 j / 2 ψ ( 2 j x − k ) , j , k ∈ Z {\psi _{j,k}}(x) = {2^{j/2}}\psi ({2^j}x - k),j,k \in \mathbb {Z} , of L 2 ( R ) {L^2}(\mathbb {R}) , we prove that { n − 1 / 2 ψ j , k / n } \{ {n^{ - 1/2}}{\psi _{j,k/n}}\} is also an affine frame of L 2 ( R ) {L^2}(\mathbb {R}) with the same frame bounds for any positive odd integer n . This establishes the result stated as the title of this paper. A counterexample of this statement for n = 2 n = 2 is also given.