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: 2008
• 资助国家: 加拿大
• 起止时间: 2008-01-01 至 2009-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=394317
• 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，最近在信号分析和处理方面有许多应用，例如语音、图像、心电图以及更通用的数字数据集。它提供了在不同尺度上解析信号和提取特征的工具，以及对存储和计算有用的信息压缩方法。一般来说，谐波分析涉及将信息分解成更简单的部分。这可以用音乐模型很好地说明。当敲击中C音叉时，它会产生近乎纯音的音调；即气压的正弦波变化。另一方面，当诸如小提琴之类的乐器演奏中音C时，它会产生纯音的总和：在同一基频或基频上以不同强度叠加的，是其频率倍数的正弦波。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性或音色。尽管两种情况下演奏的音符完全相同，但这两种乐器听起来略有不同。人耳在识别由于不同幅度的高次谐波的存在而引起的声音特征的这种微妙差异时，会进行谐波分析。我的工作主要涉及乘数理论，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，它已以这种方式分解为不同幅度的组成频率，并假设我们有兴趣阻尼其中某些频率，并可能放大其他频率：例如，当调整立体声音响上的均衡器。乘数理论研究这会对整个信号产生什么影响。乘子理论在许多数学领域都有重要的应用，例如偏微分方程、解析数论、微分几何，甚至在纳维斯托克斯问题中发挥着重要作用，该问题的解现在价值100万美元。