An Abstract Formulation of Wavelets

摘要Baggett最近，人们发现了小波的一种内部精细结构。也就是说，每个小波都与一个“多重性函数”相关联，这个“多重性函数”似乎携带着小波在频域中的离散性的一些信息。 这样的冗余应该导致更可靠的重建信号从他们的小波变换。 本研究着重于这些多重功能的理论本质。 也就是说，我们将研究这种多重性函数的性质，试图准确地确定哪种函数以这种方式出现，并尝试根据它们的多重性函数对小波进行分类。例如，信号可以从这些数字中（相对准确地）再现;它可以存储在计算机中，以后再检查;或者它本身可以被指定为“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，加上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。 这些单一的固定标准（小波）的各种已知的，但它们在信号分析中的比较优点仍在开发中。 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，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.