Some problems related to wavelet packet bases and convergence

الملخص EN

Wavelet packets are subsets of a multiresolution analysis and derive many of their properties therefrom.

Those defined by a single filter pair have uncontrolled size and basis properties, in general.

By substituting different filters at different scales according to a rule, these can be controlled.

The number of orthonormal bases available in an MRA satisfies a recursion equation depending on the basis selection method, and some of these recursions have closed form solutions.

Some of these orthonormal bases consist of uniformly bounded, uniformly compactly supported wavelet packets and are Schauder bases for many Banach spaces.

With controlled size and support, the Carleson–Hunt theorem applies to show that a wavelet packet Fourier series of a continuous function converges pointwise almost everywhere.