Interpolatory frames in signal space

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 low-pass filter and two high-pass filters, one of which is also interpolatory and symmetric. The second high-pass 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 the 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 infinite-impulse response (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.