Tight Frame Oversampling and Its Equivalence to Shift-invariance of Affine Frame Operators
نویسندگان
چکیده
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 shiftinvariance with respect to 1 mN0 Z of the affine frame operator Q0,N0 defined on the zeroth scale level.
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