A note on the standard dual frame of a wavelet frame with three-scale

In this paper, it is shown that there exist wavelet frames generated by two functions which have good dual wavelet frames, but for which the standard dual wavelet frame does not consist of wavelets. That is to say, the standard dual wavelet frame cannot be generated by the translations and dilations of a single function. Relation to some physical theories such as entropy and E-infinity theory is also discussed.