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 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 Classification
Pure or Applied Connections
Associated With
Includes
Source Bochner's Theorem (Wikipedia)
Source Wikipedia: Edwin Hewitt
Source Fejér Kernel (Wikipedia)
Source Fourier Series (Wikipedia)
Source Wikipedia: Joseph Fourier
Source Josiah Willard Gibbs, Biography (MacTutor History of Mathematics)
Source Wikipedia: Lennart Carleson
Source Lipot Fejer (MacTutor History of Mathematics)
Source Marc-Antoine Parseval (Wikipedia)
Source Wikipedia: Michel Plancherel
Source Plancherel theorem (Wikipedia)
Source Poisson summation formula (Wikipedia)
Source Terence Tao (Wikipedia)
Source Titchmarsh convolution theorem (Wikipedia)
Sources
1. Wikipedia: Harmonic Analysis
Wikimedia FoundationLead section
The methods of harmonic analysis decompose functions and related objects, such as measures, 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 Plancherel theorem (Wikipedia)
Marc-Antoine Parseval (Wikipedia)
Includes: Marc-Antoine Parseval, Lead paragraphQuote, Includes: Marc-Antoine Parseval, Lead paragraph
the Fourier transform is unitary
Wikipedia: Edwin Hewitt
Includes: Edwin Hewitt, LeadQuote, Includes: Edwin Hewitt, Lead
known for his work in abstract harmonic analysis
Wikipedia: Lennart Carleson
Includes: Lennart Carleson, LeadQuote, Includes: Lennart Carleson, Lead
known as a leader in the field of harmonic analysis
Wikipedia: Michel Plancherel
Includes: Michel Plancherel, LeadQuote, Includes: Michel Plancherel, Lead
is known for the Plancherel theorem in harmonic analysis
Wikipedia: Joseph Fourier
Includes: Joseph Fourier, LeadQuote, Includes: Joseph Fourier, Lead
best known for initiating the investigation of Fourier series, which eventually developed into Fourier analysis and harmonic analysis
Terence Tao (Wikipedia)
Includes: Terence Tao, Lead paragraphQuote, Includes: Terence Tao, Lead paragraph
He was awarded the Fields Medal in 2006 for his contributions to partial differential equations, combinatorics, harmonic analysis, and additive number theory.
Bochner's Theorem (Wikipedia)
Wikimedia FoundationIncludes: Bochner's Theorem, Lead paragraphQuote, Includes: Bochner's Theorem, Lead paragraph
More generally in harmonic analysis, Bochner's theorem asserts that under Fourier transform a continuous positive-definite function on a locally compact abelian group corresponds to a finite positive measure on the Pontryagin dual group.
Lipot Fejer (MacTutor History of Mathematics)
Includes: Lipót Fejér, BiographyQuote, Includes: Lipót Fejér, Biography
main work was in harmonic analysis
Poisson summation formula (Wikipedia)
Includes: Poisson Summation Formula, Lead sectionQuote, Includes: Poisson Summation Formula, Lead section
Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation
Fejér Kernel (Wikipedia)
Includes: Fejér Kernel, Lead sectionQuote, Includes: Fejér Kernel, Lead section
kernel used to express the effect of Cesàro summation on Fourier series. It is a non-negative kernel, giving rise
Titchmarsh convolution theorem (Wikipedia)
Includes: Titchmarsh Convolution Theorem, Lead sectionQuote, Includes: Titchmarsh Convolution Theorem, Lead section
theorem describes the properties of the support of the convolution of two functions. It was proven by Edward Charles Titchmarsh in
Josiah Willard Gibbs, Biography (MacTutor History of Mathematics)
Includes: Josiah Willard Gibbs, SummaryQuote, Includes: Josiah Willard Gibbs, Summary
J Willard Gibbs was an American mathematician best-known for the Gibbs effect seen when Fourier-analysing a discontinuous function.
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