Tight frame oversampling and its equivalence to shift-invariance of affine frame operators

Let $\Psi=\{\psi_1, \ldots, \psi_L\}\subset L^2:=L^2(-\infty, \infty)$ generate a tight affine frame with dilation factor $M$, where $2\le M\in \mathbf{Z}$, and sampling constant $b=1$ (for the zeroth scale level). Then for $1\le N\in \mathbf{Z}$, $N\times$oversampling (or oversampling by $N$) means replacing the sampling constant $1$ by $1/N$. The Second Oversampling Theorem asserts that $N\times$oversampling of the given tight affine frame generated by $\Psi$ preserves a tight affine frame, provided that $N=N_0$ is relatively prime to $M$ (i.e., $gcd(N_0,M)=1$). In this paper, we discuss the preservation of tightness in $mN_0\times$oversampling, where $1\le m\vert M$ (i.e., $1\le m\le M$and $gcd(m,M)=m$). We also show that tight affine frame preservation in $mN_0\times$oversampling is equivalent to the property of shift-invariance with respect to $\frac{1}{mN_0}\mathbf{Z}$ of the affine frame operator $Q_{0,N_0}$ defined on the zeroth scale level.