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Branches of Mathematic

Harmonic Analysis

Analysis

Harmonic analysis is the branch of mathematical analysis that studies how functions and other objects decompose into basic oscillating components, extending the reach of Fourier's original idea of representing a function as a sum of trigonometric functions. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/

Facts
Central Question
How a function or a measure can be decomposed into components organized by symmetry, scale, spectrum or oscillation, and what that decomposition reveals about the object's finer behavior. 1
Key Debate
How far Fourier's original insight, that representing a function as a sum of trigonometric functions simplifies the study of heat transfer, could be extended beyond the circle and the line to general groups and spaces, a generalization that turned a technique for one equation into a branch of analysis in its own right. 1
Classification
Pure or Applied
Pure Mathematics 1
Connections

Associated With

Functional Analysis, Branches of Mathematics

Includes

Fejer's Theorem, Theorems
Fourier Series, Concepts
Source Fourier Series (Wikipedia)
Marc-Antoine Parseval, Mathematicians
Source Marc-Antoine Parseval (Wikipedia)
Plancherel Theorem, Theorems
Source Plancherel theorem (Wikipedia)
Sources
1. Wikipedia: Harmonic Analysis
Wikimedia Foundation
  • Overview section
    The methods of harmonic analysis decompose functions and related objects into components based on symmetries, scales, spectra, or oscillation.
  • Overview section, decomposition statement
    The methods of harmonic analysis decompose functions and related objects, such as measures, into components based on symmetries, scales, spectra, or oscillation.
  • Abstract harmonic analysis section
    Abstract harmonic analysis extends the classical spherical harmonic decompositions to functions on spaces associated to other groups.
View the Source
Stone-Weierstrass Theorem (Wikipedia)
Wikipedia
  • lead paragraph
    Harmonic analysis is an area of mathematical analysis that emerged from the study of harmonic functions, and especially their boundary behavior.
  • lead section
    Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer.
View the Source
Plancherel theorem (Wikipedia)
Includes: Plancherel Theorem, Lead sentenceView the Source
Marc-Antoine Parseval (Wikipedia)
Includes: Marc-Antoine Parseval, Lead paragraph
Quote, Includes: Marc-Antoine Parseval, Lead paragraph
the Fourier transform is unitary
View the Source
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