 
Summary: Interpolatory frames in signal space
Amir Z. Averbuch Valery A Zheludev Tamir Cohen
School of Computer Science, Tel Aviv University
69978, Tel Aviv, Israel
Abstract
We present a new family of frames, which are generated by perfect reconstruction filter banks of
linear phased filters. The filter banks are based on discrete interpolatory splines and are related to
Butterworth filters. Each filter bank contains one interpolatory symmetric lowpass filter and two
highpass filters, one of which is also interpolatory and symmetric. The second highpass filter is
either symmetric or antisymmetric. These filter banks generate the analysis and synthesis scaling
functions and pairs of framelets. We introduce the concept of semitight frame. All the analysis
waveforms in a tight frame coincide with their synthesis counterparts. In the semitight frame we
can trade properties of smoothness and number of vanishing moments between the synthesis and
the analysis framelets. We construct dual pairs of frames, where all the waveforms are symmetric
and all the framelets have the same number of vanishing moments. Although most of the designed
filters are IIR, they allow fast implementation via recursive procedures. The waveforms are well
localized in time domain despite their infinite support. The frequency response of the designed
filters is flat.
Introduction
Recently frames or redundant expansions of signals have attracted considerable interest from re
