Pairs of Dual Wavelet Frames from Any Two Refinable Functions

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Compactly supported tight wavelet frames and orthonormal wavelets of exponential decay with a general dilation matrix (2003, background): “Recently, interesting progresses have been made in [3, 10, 11 ] to obtain univariate tight wavelet frames with high vanishing moments.”
Framelets: MRA-based constructions of wavelet frames☆☆☆ (2003, background): “In [17] it is shown that one can, in fact, construct bi-framelets from any pair of refinable functions φ, φ(with compact support).”
The double-density dual-tree DWT (2004, background): “We propose, for the double-density dual-tree DWT, that the set of filters take the following form: (17) (18) (19) (20) (21) (22) Equivalently (23) (24) (25) (26) (27) (28) where These expressions incorporate the moment properties.”
Dual Wavelet Frames and Riesz Bases in Sobolev Spaces (2009, background): “Proof Since φ and η are compactly supported, by (1.9) and Proposition 2.6 ,w e have μ2(φ) > s and μ2(η) > −s. By the same proof of Theorem 2.4, it follows from Proposition 2.3 and (1.10) that X s (φ; ψ 1 ,...,ψ L ) is a wavelet frame in H s (R d ) and X −s (η; ˜ ψ 1 ,..., ˜ ψ L ) is a wavelet frame ”