$n\times$ oversampling preserves any tight affine frame for odd $n$

Abstract:

If $\psi$ generates an affine frame ${\psi _{j,k}}(x) = {2^{j/2}}\psi ({2^j}x - k),j,k \in \mathbb {Z}$, of ${L^2}(\mathbb {R})$, we prove that $\{ {n^{ - 1/2}}{\psi _{j,k/n}}\}$ is also an affine frame of ${L^2}(\mathbb {R})$ 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 $n = 2$ is also given.