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Walsh-Type Wavelet Packet Expansions

✍ Scribed by Morten Nielsen

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
168 KB
Volume
9
Category
Article
ISSN
1063-5203

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✦ Synopsis

We consider a family of basic nonstationary wavelet packets generated using the Haar filters except for a finite number of scales where we allow the use of arbitrary filters. Such a system, which we call a system of Walsh-type wavelet packets, can be considered as a smooth generalization of the Walsh functions. We show that the basic Walsh-type wavelet packets share a number of metric properties with the Walsh system. We prove that the system constitutes a Schauder basis for L p (R), 1 < p < ∞, and we construct an explicit function in L 1 (R) for which the expansion fails. Then we prove that expansions of L p (R)-functions, 1 < p < ∞, in the Walshtype wavelet packets converge pointwise a.e. Finally, we prove that the analogous results are true for periodic Walsh-type wavelet packets in L p [0, 1).

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