An Abstract Formulation of Wavelets

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AbstractBaggettRecently, a kind of internal fine structure of wavelets has been discovered. That is, each wavelethas associated to it a "multiplicity function" that appears to carry some information about theredundancy, in the frequency domain, of the wavelet. Such redundancies should lead to morereliable reconstructions of signals from their wavelet transforms. This research focuses on thetheoretical nature of these multiplicity functions. That is, we will study the properties of thismultiplicity function, try to determine exactly which kinds of functions occur in this way, andattempt to classify wavelets according to their multiplicity functions.The way in which an unknown signal (radio wave, seismic shock, astronomic vibration,etc.) is typically analyzed is by numerically comparing it to a set of fixed standard signals.For instance, the signal can be reproduced (relatively accurately) from these numbers; it can be stored in a computer and re-examined later; or it can be designated as a "new" standard itself. The new idea that comes from Wavelet Theory is that this important set of fixed standards can be described in terms of a single standard, together with several expansions and shifts of it. This simplifies greatly the technology for making the comparisons with an arbitrary signal. Various of these single fixed standards (wavelets) are known, but their comparative virtues in signal analysis are still being developed. This research project deals with a newly discovered property of wavelets. Each one has associated to it a notion of "redundancy" of frequencies. It is thought that these redundancies might be an important feature of the wavelet, one that could further simplify and improve the accuracy and reliability of signal analysis.