# Connectivity, homotopy degree, and other properties of α-localized wavelets on R.

Publicacions Matemàtiques (1999)

- Volume: 43, Issue: 1, page 303-340
- ISSN: 0214-1493

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topGarrigós, Gustavo. "Connectivity, homotopy degree, and other properties of α-localized wavelets on R.." Publicacions Matemàtiques 43.1 (1999): 303-340. <http://eudml.org/doc/41356>.

@article{Garrigós1999,

abstract = {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.},

author = {Garrigós, Gustavo},

journal = {Publicacions Matemàtiques},

keywords = {Espacio de medida; Topología algebraica; Homotopía; Ondículas; phases; homotopy degree; -localized wavelets; multiresolution analyses; low-pass filters; Fourier transforms; scaling functions; MRA},

language = {eng},

number = {1},

pages = {303-340},

title = {Connectivity, homotopy degree, and other properties of α-localized wavelets on R.},

url = {http://eudml.org/doc/41356},

volume = {43},

year = {1999},

}

TY - JOUR

AU - Garrigós, Gustavo

TI - Connectivity, homotopy degree, and other properties of α-localized wavelets on R.

JO - Publicacions Matemàtiques

PY - 1999

VL - 43

IS - 1

SP - 303

EP - 340

AB - 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.

LA - eng

KW - Espacio de medida; Topología algebraica; Homotopía; Ondículas; phases; homotopy degree; -localized wavelets; multiresolution analyses; low-pass filters; Fourier transforms; scaling functions; MRA

UR - http://eudml.org/doc/41356

ER -

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