Abstract
Let A be a d×d expansive matrix with |detA|=2. An A-wavelet is a function \(\psi\in L^{2}(\mathbb{R}^{d})\) such that \(\{2^{\frac{j}{2}}\psi(A\cdot-k):\,j\in \mathbb{Z},\,k\in \mathbb{Z}^{d}\}\) is an orthonormal basis for \(L^{2}(\mathbb{R}^{d})\). A measurable function f is called an A-wavelet multiplier if the inverse Fourier transform of \(f\hat{\psi}\) is an A-wavelet whenever ψ is an A-wavelet, where \(\hat{\psi}\) denotes the Fourier transform of ψ. A-scaling function multiplier, A-PFW multiplier, semi-orthogonal A-PFW multiplier, MRA A-wavelet multiplier, MRA A-PFW multiplier and semi-orthogonal MRA A-PFW multiplier are defined similarly. In this paper, we prove that the above seven classes of multipliers are equivalent, and obtain a characterization of them. We then prove that if the set of all A-wavelet multipliers acts on some A-scaling function (A-wavelet, A-PFW, semi-orthogonal A-PFW, MRA A-wavelet, MRA A-PFW, semi-orthogonal MRA A-PFW), the orbit is arcwise connected in \(L^{2}(\mathbb{R}^{d})\), and that if the generator of an orbit is an MRA A-PFW, the orbit is equal to the set of all MRA A-PFWs whose Fourier transforms have same module, and is also equal to the set of all MRA A-PFWs with corresponding pseudo-scaling functions having the same module of their Fourier transforms.
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The authors would like to thank the referees for their valuable suggestions, which greatly improve the readability of this article.
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Communicated by David Walnut.
Supported by the National Natural Science Foundation of China (Grant No. 11271037), Beijing Natural Science Foundation (Grant No. 1122008) and the Scientific Research Common Program of Beijing Municipal Commission of Education (Grant No. KM201110005030).
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Li, YZ., Xue, YQ. The Equivalence Between Seven Classes of Wavelet Multipliers and Arcwise Connectivity They Induce. J Fourier Anal Appl 19, 1060–1077 (2013). https://doi.org/10.1007/s00041-013-9282-5
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DOI: https://doi.org/10.1007/s00041-013-9282-5