Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

Completeness in the set of wavelets

Author(s): Gustavo Garrigós; Darrin Speegle
Journal: Proc. Amer. Math. Soc. 128 (2000), 1157-1166.
MSC (1991): Primary 42C15
Posted: August 17, 1999
MathSciNet review: 1646304
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Abstract: We study the completeness properties of the set of wavelets in $L^{2}(\mathbb{R})$ . It is well-known that this set is not closed in the unit ball of $L^{2}(\mathbb{R})$ . However, if one considers the metric inherited as a subspace (in the Fourier transform side) of $L^{2}(\mathbb{R},d\xi ) \cap L^{2}(\mathbb{R}_{*},{\frac{{d\xi }}{{|\xi |}}})$ , we do obtain a complete metric space.

References:

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[HKLS]: Y-H. Ha, H. Kang, J. Lee and J. Seo, Unimodular wavelets for $L^{2}$ and the Hardy space $H^{2}$ , Michigan Math. J. 41 (1994), 345-361. MR 95g:42050
[HW]: Hernández, E. and Weiss, G. L., A First Course on Wavelets, CRC Press, 1996. MR 97i:42015
[SPEE]: D. M. Speegle, The s-elementary wavelets are path connected, Proc. AMS 127 (1999), no. 1, 223-233. MR 99b:42045

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