The Equivalence Between Seven Classes of Wavelet Multipliers and Arcwise Connectivity They Induce

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, MRAA-wavelet multiplier, MRAA-PFW multiplier and semi-orthogonal MRAA-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, MRAA-wavelet, MRAA-PFW, semi-orthogonal MRAA-PFW), the orbit is arcwise connected in \(L^{2}(\mathbb{R}^{d})\), and that if the generator of an orbit is an MRAA-PFW, the orbit is equal to the set of all MRAA-PFWs whose Fourier transforms have same module, and is also equal to the set of all MRAA-PFWs with corresponding pseudo-scaling functions having the same module of their Fourier transforms.