On the Density Condition of a Multiresolution Analysis in Lebesgue Spaces

We are interested in the problem of when the density condition in a multiresolution analysis defined in L p ( R n ) , 1 ≤ p ≤ ∞ , holds. Indeed, if 2 ≤ p < ∞ , we obtain sufficient conditions on the generators of a multiresolution analysis in order to the density condition is satisfied. We emphas...

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Bibliographic Details
Published inMediterranean journal of mathematics Vol. 14; no. 3; pp. 1 - 13
Main Author Antolín, A. San
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.06.2017
Springer Nature B.V
Subjects
Continuity (mathematics)
Fourier analysis
Fourier transforms
Generators
Mathematics
Mathematics and Statistics
Multiresolution analysis
Residential density
43A15
point of density
46F12
42A65
42B10
Lebesgue spaces
Fourier transform
multiresolution analysis
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Summary:We are interested in the problem of when the density condition in a multiresolution analysis defined in L p ( R n ) , 1 ≤ p ≤ ∞ , holds. Indeed, if 2 ≤ p < ∞ , we obtain sufficient conditions on the generators of a multiresolution analysis in order to the density condition is satisfied. We emphasis on the requirement of the Fourier transform in a neighborhood of the origin. This involves the notion of density point. When 1 ≤ p ≤ 2 , the obtained condition is necessary. Moreover, we study the same problem when a multiresolution analysis is defined in the subspace of L ∞ ( R n ) of the set of all continuous functions vanishing at infinite.
ISSN:1660-5446
1660-5454
DOI:10.1007/s00009-017-0908-8