Overview and Goals Data arriving in time order (a data stream) arises in fields ranging from physics to finance to medicine to music, just to name a few. Often the data comes from sensors (in physics and medicine for example) whose data rates continue to improve dramati cally as sensor technology improves. Further, the number of sensors is increasing, so correlating data between sensors becomes ever more critical in orderto distill knowl edge from the data. On-line response is desirable in many applications (e.g., to aim a telescope at a burst of activity in a galaxy or to perform magnetic resonance-based real-time surgery). These factors - data size, bursts, correlation, and fast response motivate this book. Our goal is to help you design fast, scalable algorithms for the analysis of single or multiple time series. Not only will you find useful techniques and systems built from simple primi tives, but creative readers will find many other applications of these primitives and may see how to create new ones of their own. Our goal, then, is to help research mathematicians and computer scientists find new algorithms and to help working scientists and financial mathematicians design better, faster software."

Harmonic Analysis is an important tool that plays a vital role in many areas of mathematics as well as applications. It studies functions by decomposing them into components that are special functions. A prime example is decomposing a periodic function into a linear combination of sines and cosines. The subject is vast, and this book covers only the selection of topics that was dealt with in the course given at the Courant Institute in 2000 and 2019. These include standard topics like Fourier series and Fourier transforms of functions, as well as issues of convergence of Abel, Feier, and Poisson sums. At a slightly more advanced level the book studies convolutions with singular integrals, fractional derivatives, Sobolev spaces, embedding theorems, Hardy spaces, and BMO. Applications to elliptic partial differential equations and prediction theory are explored. Some space is devoted to harmonic analysis on compact non-Abelian groups and their representations, including some details about two groups: the permutation group and SO(3). The text contains exercises at the end of most chapters and is suitable for advanced undergraduate students as well as first- or second-year graduate students specializing in the areas of analysis, PDE, probability or applied mathematics.

This monograph is a technical survey of concepts and techniques for describing and analyzing large-scale time-series data streams. Some topics covered are algorithms for query by humming, gamma-ray burst detection, pairs trading, and density detection. Included are self-contained descriptions of wavelets, fast Fourier transforms, and sketches as they apply to time-series analysis. Detailed applications are built on a solid scientific basis.