From Filters to Wavelets via Direct Limits II: Wavelet-Packet Bases

In recent joint work with Nadia Larsen, we gave a new proof of a theorem of Mallat which describes how to construct wavelets from quadrature mirror filters. Our main innovation was to show how the scaling function associated to the filter can be used to identify a particular direct limit of Hilbert spaces with L ²(ℝ). Here we show that wavelet-packet bases for L ²(ℝ) also fit naturally into the same direct-limit framework.