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Table of contents
The theoretical approach is often to try to describe the system by a differential equation or system of equations to predict the essential features, including the amplitude, frequency, and phases of the oscillatory components. The specific equations depend on the field, but theories generally try to select equations that represent major principles that are applicable.
The experimental approach is usually to acquire data that accurately quantifies the phenomenon. For example, in a study of tides, the experimentalist would acquire samples of water depth as a function of time at closely enough spaced intervals to see each oscillation and over a long enough duration that multiple oscillatory periods are likely included. In a study on vibrating strings, it is common for the experimentalist to acquire a sound waveform sampled at a rate at least twice that of the highest frequency expected and for a duration many times the period of the lowest frequency expected.
The waveform appears oscillatory, but it is more complex than a simple sine wave, indicating the presence of additional waves. The different wave components contributing to the sound can be revealed by applying a mathematical analysis technique known as the Fourier transform , the result of which is shown in the lower figure.
One of the most modern branches of harmonic analysis, having its roots in the midth century, is analysis on topological groups.
The core motivating ideas are the various Fourier transforms , which can be generalized to a transform of functions defined on Hausdorff locally compact topological groups. The theory for abelian locally compact groups is called Pontryagin duality. Harmonic analysis studies the properties of that duality and Fourier transform and attempts to extend those features to different settings, for instance, to the case of non-abelian Lie groups.
For general non-abelian locally compact groups, harmonic analysis is closely related to the theory of unitary group representations. For compact groups, the Peter—Weyl theorem explains how one may get harmonics by choosing one irreducible representation out of each equivalence class of representations. This choice of harmonics enjoys some of the useful properties of the classical Fourier transform in terms of carrying convolutions to pointwise products, or otherwise showing a certain understanding of the underlying group structure.
If the group is neither abelian nor compact, no general satisfactory theory is currently known "satisfactory" means at least as strong as the Plancherel theorem. However, many specific cases have been analyzed, for example SL n. In this case, representations in infinite dimensions play a crucial role.
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From Wikipedia, the free encyclopedia. For the process of determining the structure of a piece of music, see Harmony. Product details Hardcover Publisher: View shipping rates and policies Average Customer Review: Be the first to review this item Would you like to tell us about a lower price? Related Video Shorts 0 Upload your video.
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