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

Abstract

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 waveletpacket bases for L 2 (R) also fit naturally into the same direct-limit framework.Keywords Wavelet · Multiresolution analysis · Wavelet packet · Cuntz family · Direct limit · Hilbert space A famous theorem of Mallat [5] describes how to construct wavelets from a quadrature mirror filter, which is a function m 0 : T → C satisfying |m 0 (z)| 2 + |m 0 (−z)| 2 = 1 for every z ∈ T.At the meeting in Banff, I discussed an approach to Mallat's theorem arising from joint work with Nadia Larsen [3]. The main ingredients in our proof are a pair of isometries S i on L 2 (T) satisfying the Cuntz relation S 0 S * 0 + S 1 S * 1 = 1, and a direct limit of Hilbert spaces associated to S 0 . After my talk, in response to a question about possible variations to our construction, Palle Jorgensen and Karlheinz Gröchenig suggested that I look at wavelet packets. When I followed up their suggestion, I found that wavelet-packet bases also fit very naturally into the direct-limit framework proposed in [3], though maybe not quite the way Palle and Karlheinz expected. 1 My goal in this sequel is to explain this new application of the construction from [3].