An Abstract Formulation of Wavelets

摘要近年来，人们发现了小波的一种内部精细结构。也就是说，每个小波都有一个与之相关的“多重函数”，它似乎携带着小波在频域的冗余信息。这样的冗余将导致更可靠的小波变换信号重建。本研究的重点是这些多重函数的理论性质。也就是说，我们将研究这个多重函数的性质，试图确定哪种函数会以这种方式出现，并尝试根据它们的多重函数对小波进行分类。分析未知信号（无线电波、地震冲击、天文振动等）的典型方法是将其与一组固定的标准信号进行数值比较。例如，信号可以（相对准确地）从这些数字中再现；它可以储存在电脑里，以后再检查；或者它本身也可以被指定为一个“新”标准。来自小波理论的新思想是，这组重要的固定标准可以用一个标准来描述，连同它的几个扩展和移位。这大大简化了与任意信号进行比较的技术。各种单一的固定标准（小波）是已知的，但它们在信号分析中的相对优点仍在开发中。这个研究项目涉及小波的一个新发现的性质。每一个都与频率的“冗余”概念相关联。人们认为这些冗余可能是小波的一个重要特征，它可以进一步简化和提高信号分析的准确性和可靠性。

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.