Please use this identifier to cite or link to this item: https://doi.org/10.1090/S0002-9939-02-06703-5
Title: Tight frame oversampling and its equivalence to shift-invariance of affine frame operators
Authors: Chui, C.K.
Sun, Q. 
Issue Date: May-2003
Citation: Chui, C.K., Sun, Q. (2003-05). Tight frame oversampling and its equivalence to shift-invariance of affine frame operators. Proceedings of the American Mathematical Society 131 (5) : 1527-1538. ScholarBank@NUS Repository. https://doi.org/10.1090/S0002-9939-02-06703-5
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.
Source Title: Proceedings of the American Mathematical Society
URI: http://scholarbank.nus.edu.sg/handle/10635/104646
ISSN: 00029939
DOI: 10.1090/S0002-9939-02-06703-5
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