Revistes Catalanes amb Accés Obert (RACO)

Connectivity, homotopy degree, and other properties of $\alpha$-localized wavelets on $\Bbb R$

G. Garrigós


In this paper, we study general properties of $\alpha$-localized wavelets and multiresolution analyses, when $\frac12 < \alpha\leq\infty$. Related to the latter, we improve a well-known result of A. Cohen by showing that the correspondence $m\mapsto{\widehat\varphi}=\prod_1^\infty m(2^{-j}\cdot)$, between low-pass filters in $H^{\alpha}(\Bbb T)$ and Fourier transforms of $\alpha$-localized scaling functions (in $H^{\alpha}(\Bbb R)$), is actually a homeomorphism of topological spaces. We also show that the space of such filters can be regarded as a connected infinite dimensional manifold, extending a theorem of A. Bonami, S. Durand and G. Weiss, in which only the case $\alpha=\infty$ is treated. These two properties, together with a careful study of the "phases" that give rise to a wavelet from the MRA, will allow us to prove that the space ${\Cal W}_\alpha$, of $\alpha$-localized wavelets, is arcwise connected with the topology of $L^2((1+|x|^2)^{\alpha}\,dx)$ (modulo homotopy classes). This last result is new even for the case $\alpha=\infty$, as well as the considerations about the "homotopy degree" of a wavelet.

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