Branches of Mathematics
Harmonic Analysis
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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.
Facts
Central QuestionHow 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 DebateHow 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 Cross-Tradition Connections
Sources
1. Wikipedia: Harmonic Analysis
Wikimedia FoundationOverview sectionQuote, Overview section
The methods of harmonic analysis decompose functions and related objects into components based on symmetries, scales, spectra, or oscillation.
1. Wikipedia: Harmonic Analysis
Wikimedia FoundationOverview section, decomposition statementQuote, 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.
1. Wikipedia: Harmonic Analysis
Wikimedia FoundationAbstract harmonic analysis sectionQuote, Abstract harmonic analysis section
Abstract harmonic analysis extends the classical spherical harmonic decompositions to functions on spaces associated to other groups.
Stone-Weierstrass Theorem (Wikipedia)
Wikipedialead paragraphQuote, lead paragraph
Harmonic analysis is an area of mathematical analysis that emerged from the study of harmonic functions, and especially their boundary behavior.
Stone-Weierstrass Theorem (Wikipedia)
Wikipedialead sectionQuote, 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.
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