Abstract
We are interested in the problem of when the density condition in a multiresolution analysis defined in \(L^p({\mathbb {R}}^n)\), \(1 \le p \le \infty \), holds. Indeed, if \(2 \le p < \infty \), 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 \(1 \le p \le 2\), the obtained condition is necessary. Moreover, we study the same problem when a multiresolution analysis is defined in the subspace of \(L^{\infty }({\mathbb {R}}^n)\) of the set of all continuous functions vanishing at infinite.
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We would like to thank the anonymous referee for his useful comments and suggestions, which provided an improvement in the overall presentation and clarity of this manuscript.
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The author was partially supported by MEC/MICINN Grant #MTM2011-27998 (Spain) and by Generalitat Valenciana Grant GV/2015/035.
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Antolín, A.S. On the Density Condition of a Multiresolution Analysis in Lebesgue Spaces. Mediterr. J. Math. 14, 106 (2017). https://doi.org/10.1007/s00009-017-0908-8
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DOI: https://doi.org/10.1007/s00009-017-0908-8