Abstract
Let A be any 2×2 real expansive matrix. For any A-dilation wavelet ψ, let \(\widehat{\psi}\) be its Fourier transform. A measurable function f is called an A-dilation wavelet multiplier if the inverse Fourier transform of \((f\widehat{\psi})\) is an A-dilation wavelet for any A-dilation wavelet ψ. In this paper, we give a complete characterization of all A-dilation wavelet multipliers under the condition that A is a 2×2 matrix with integer entries and |{det }(A)|=2. Using this result, we are able to characterize the phases of A-dilation wavelets and prove that the set of all A-dilation MRA wavelets is path-connected under the L 2(ℝ2) norm topology for any such matrix A.
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Communicated by Yang Wang.
Zhongyan Li is supported by the grant of Young Teachers Study Abroad of China Scholarship Council (2005) and Yuanan Diao is partially supported by NSF grant DMS-0712958.
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Li, Z., Dai, X., Diao, Y. et al. Multipliers, Phases and Connectivity of MRA Wavelets in L 2(ℝ2). J Fourier Anal Appl 16, 155–176 (2010). https://doi.org/10.1007/s00041-009-9089-6
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DOI: https://doi.org/10.1007/s00041-009-9089-6