Connectivity, homotopy degree, and other properties of [alpha]-localized wavelets on R
Garrigós, Gustavo

Data: 1999
Resum: In this paper, we study general properties of α-localized wavelets and multiresolution analyses, when 1 < α ≤ ∞. Related to the 2 latter, we improve a well-known result of A. Cohen by showing ∞ that the correspondence m → ϕ = 1m(2−j ·), between lowpass filters in H α (T) and Fourier transforms of α-localized scaling functions (in H α (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 + $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.
Drets: Tots els drets reservats.
Llengua: Anglès.
Document: article ; recerca ; publishedVersion
Publicat a: Publicacions matematiques, V. 43 N. 1 (1999) , p. 303-340, ISSN 0214-1493

Adreça original:
DOI: 10.5565/PUBLMAT_43199_14

38 p, 289.3 KB

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