Multiparameter harmonic analysis on the Heisenberg group

海森堡群的多参数调和分析

• 批准号: 240545-2006
• 负责人: Fraser, Andrea
• 金额: $ 0.44万
• 依托单位: Dalhousie University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2006
• 资助国家: 加拿大
• 起止时间: 2006-01-01 至 2007-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=339672
• 关键词: Multiparameter; harmonic; analysis; Heisenberg; group

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.

谐波分析是一个高度活跃的数学领域，它在分析和操纵语音，图像，心电图以及更一般的数字数据集等信号方面已有许多最新应用。它提供了用于阐明信号和在不同尺度上提取功能的工具，以及压缩信息的方法，这些信息可用于存储和计算。一般而言，谐波分析涉及将信息分解为简单的部分。可以通过音乐模型很好地说明这一点。当撞击中间C的调谐叉时，它几乎会产生纯正的音调。也就是说，气压中的正弦波变化。另一方面，当小提琴之类的乐器扮演中间C时，它会产生一定的纯音：在同一地面或基本频率上以各种强度叠加的是带有频率的正弦波，这是倍数的。这些称为较高的谐波。这些较高的谐波的存在负责仪器的性格或音色。尽管在两种情况下播放了完全相同的音符，但两种乐器的听起来略有不同。在认识到由这些较高谐波在各种振幅下存在的声音特征的这种微妙差异时，人的耳朵正在进行谐波分析。我的工作在很大程度上涉及乘数理论，这是谐波分析中非常重要的工具。可以在音乐模型中非常描述它。如果我们考虑以这种方式分解为各种幅度的构成频率的声音或信号，并假设我们有兴趣抑制某些频率，也许会放大其他频率：例如，在立体声集中调整均衡器时，我们会这样做。乘数理论是对这对整个信号的影响的研究。乘数理论在许多数学领域（例如偏微分方程，分析数理论，差异几何形状，甚至在Navier Stokes问题中起作用）中具有重要的应用，该解决方案现在值得我们\ 100万美元。