In this paper, we study general properties of α-localized wavelets and multiresolution analyses, when 1/2 < α ≤ ∞. Related to the latter, we improve a well-known result of A. Cohen by showing that the correspondence m → φ' = Π1 ∞ m(2−j ·), between low-pass filters in Hα(T) and Fourier transforms of α-localized scaling functions (in Hα(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 α = ∞ 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 Wα, of α-localized wavelets, is arcwise connected with the topology of L2((1 + |x|2)α dx) (modulo homotopy classes). This last result is new even for the case α = ∞, as well as the considerations about the “homotopy degree” of a wavelet.
@article{urn:eudml:doc:41356,
title = {Connectivity, homotopy degree, and other properties of $\alpha$-localized wavelets on R.},
journal = {Publicacions Matem\`atiques},
volume = {43},
year = {1999},
pages = {303-340},
mrnumber = {MR1697527},
zbl = {0933.42024},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41356}
}
Garrigós, Gustavo. Connectivity, homotopy degree, and other properties of α-localized wavelets on R.. Publicacions Matemàtiques, Tome 43 (1999) pp. 303-340. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41356/