Multiparameter harmonic analysis on the Heisenberg group

谐波分析是一个非常活跃的数学领域，它最近在语音、图像、心电以及更一般的数字数据集的分析和处理中得到了许多应用。它提供了分解信号和提取不同尺度的特征的工具，以及压缩对存储和计算有用的信息的方法。一般来说，调和分析关注的是将信息分解成更简单的片段。这可以用音乐的模型很好地说明。当中C音叉被敲击时，它会产生近乎纯净的音调，也就是说，气压的正弦波变化。另一方面，当小提琴等乐器演奏中C时，它会产生一系列纯音：以不同强度叠加在同一基频或基频上的正弦波，其频率是正弦波的倍数。这些被称为高次谐波。正是这些高次谐波的存在决定了乐器的特征或音色。尽管在这两种情况下演奏的是完全相同的音符，但两种乐器的声音略有不同。人类的耳朵在识别由不同幅度的高次谐波引起的声音特征上的细微差别时，正在进行谐波分析。我的工作主要涉及乘子理论，它是调和分析中非常重要的工具。在音乐模型中可以非常简单地描述它。如果我们考虑一些声音或信号，这些声音或信号以这种方式分解成不同幅度的组成频率，假设我们对衰减其中某些频率感兴趣，或者放大其他频率：例如，当调整立体声设备的均衡器时，我们就这样做。乘数理论是研究这将对整个信号产生什么影响的理论。乘子理论在许多数学领域都有重要的应用，如偏微分方程组、解析数论、微分几何，甚至在Navier-Stokes问题中扮演着重要的角色，该问题的解现在价值100万美元。

Harmonic analysis is a highly active field of mathematics, which has had many recent applications to the analysis and manipulation of signals such as speech, images, electrocardiograms, as well as more general digital data sets. It provides tools for unraveling signals and extracting features at different scales, as well as methods of compressing information which are useful for storage and computation. Harmonic analysis is concerned, generally speaking, with breaking information up into simpler pieces. This can be nicely illustrated with the model of music. When a tuning fork for middle C is struck, it produces nearly a pure tone; that is, a sine wave variation in the air pressure. On the other hand, when a musical instrument such as a violin plays middle C, it produces a sum of pure tones: superimposed at various intensities on the same ground or fundamental frequency, are sine waves with frequencies which are multiples of this. These are called higher harmonics. It is the presence of these higher harmonics which is responsible for the character, or timbre, of an instrument. Although exactly the same note is being played in the two cases, the two instruments sound slightly different. The human ear, in recognizing this subtle difference in the character of the sound caused by the presence of these higher harmonics at various amplitudes, is doing harmonic analysis. My work largely concerns multiplier theory, which is a very important tool in harmonic analysis. It can be described very simply in the musical model. If we consider some sound, or signal, which has been decomposed in this way into constituent frequencies at various amplitudes, and suppose we are interested in damping certain of those frequencies, and perhaps magnifying others: we do this, for example, when adjusting the equalizer on a stereo set. Multiplier theory is the study of what effect this would have on the signal as a whole. Multiplier theory has important applications in many areas of mathematics, such as partial differential equations, analytic number theory, differential geometry, and even plays a role in the Navier Stokes problem, the solution of which is now worth US\$1m.