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

Abstract

Let Ψ = {ψ1,..., ψL} ⊂ L2 := L2(-∞, ∞) generate a tight affine frame with dilation factor M, where 2 ≤ M ∈ Z, and sampling constant b = 1 (for the zeroth scale level). Then for 1 ≤ N ∈ Z, N × oversampling (or oversampling by N) means replacing the sampling constant 1 by 1/N. The Second Oversampling Theorem asserts that N × oversampling of the given tight affine frame generated by Ψ preserves a tight affine frame, provided that N = N0 is relatively prime to M (i.e., gcd(N0, M) = 1). In this paper, we discuss the preservation of tightness in mN0 × oversampling, where 1 ≤ m|M (i.e., 1 ≤ m ≤ M and gcd(m, M) = m). We also show that tight affine frame preservation in mN0 × oversampling is equivalent to the property of shift-invariance with respect to 1/mN0 of the affine frame operator Q0,N0 defined on the zeroth scale level.