391:
399:
554:
439:, shown in the lower figure. There is a prominent peak at 55 Hz, but other peaks at 110 Hz, 165 Hz, and at other frequencies corresponding to integer multiples of 55 Hz. In this case, 55 Hz is identified as the fundamental frequency of the string vibration, and the integer multiples are known as
434:
For example, the top signal at the right is a sound waveform of a bass guitar playing an open string corresponding to an A note with a fundamental frequency of 55 Hz. The waveform appears oscillatory, but it is more complex than a simple sine wave, indicating the presence of additional waves.
349:
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 valuable properties of the classical
Fourier transform in terms of carrying convolutions to pointwise products or otherwise
430:
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
422:
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 significant principles that are applicable.
431:
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.
221:(these include functions of compact support), then its Fourier transform is never compactly supported (i.e., if a signal is limited in one domain, it is unlimited in the other). This is an elementary form of an
491:
that have no analog on general groups. For example, the fact that the
Fourier transform is rotation-invariant. Decomposing the Fourier transform into its radial and spherical components leads to topics such as
406:
Many applications of harmonic analysis in science and engineering begin with the idea or hypothesis that a phenomenon or signal is composed of a sum of individual oscillatory components. Ocean
513:
are generalized harmonic functions, with respect to a symmetry group. They are an old and at the same time active area of development in harmonic analysis due to their connections to the
972:
270:
Abstract harmonic analysis is primarily concerned with how real or complex-valued functions (often on very general domains) can be studied using symmetries such as
70:, although the term is sometimes used interchangeably with harmonic analysis. Harmonic analysis has become a vast subject with applications in areas as diverse as
345:
Harmonic analysis is closely related to the theory of unitary group representations for general non-abelian locally compact groups. For compact groups, the
1036:
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If the group is neither abelian nor compact, no general satisfactory theory is currently known ("satisfactory" means at least as strong as the
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The different wave components contributing to the sound can be revealed by applying a mathematical analysis technique known as the
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and its relatives); this field is of course related to real-variable harmonic analysis, but is perhaps closer in spirit to
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is still an area of ongoing research, particularly concerning
Fourier transformation on more general objects such as
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241:
236:. There are four versions of the Fourier transform, dependent on the spaces that are mapped by the transformation:
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945:. (Introduces the decomposition of functions into odd + even parts as a harmonic decomposition over
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Harmonic
Analysis on Symmetric Spaces-Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane
331:
attempts to extend those features to different settings, for instance, first to the case of general
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One of the most modern branches of harmonic analysis, having its roots in the mid-20th century, is
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are common and simple examples. The theoretical approach often tries to describe the system by a
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981:. Translated from the 1985 Russian-language edition (Kharkov, Ukraine). Birkhäuser Verlag. 1988.
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327:. Harmonic analysis studies the properties of that duality. Different generalization of
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696:"Harmonic analysis | Mathematics, Fourier Series & Waveforms | Britannica"
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Mathematical
Framework for Pseudo-Spectra of Linear Stochastic Difference Equations
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Harmonic
Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals
938:
879:
504:
457:
189:
39:
763:
Introduction to the
Representation Theory of Compact and Locally Compact Groups
733:
Introduction to the
Representation Theory of Compact and Locally Compact Groups
503:
Harmonic analysis on tube domains is concerned with generalizing properties of
402:
Fourier transform of bass-guitar time signal of open-string A note (55 Hz)
201:, we can attempt to translate these requirements into the Fourier transform of
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for functions on bounded domains, especially periodic functions on finite
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20:
19:
For the process of determining the structure of a piece of music, see
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Harmonic analysis as the exploitation of symmetry–a historical survey
66:. Generalizing these transforms to other domains is generally called
209:
is an example. The Paley–Wiener theorem immediately implies that if
121:. Still, the term has been generalized beyond its original meaning.
483:
Harmonic analysis on
Euclidean spaces deals with properties of the
915:
Topics in
Harmonic Analysis Related to the Littlewood-Paley Theory
365:). However, many specific cases have been analyzed, for example,
197:. For instance, if we impose some requirements on a distribution
407:
1018:
655:
https://www.math.ru.nl/~burtscher/lecturenotes/2021PDEnotes.pdf
979:
Introduction to the Theory of Banach Representations of Groups
548:
476:
is also considered a branch of harmonic analysis. See, e.g.,
228:
Fourier series can be conveniently studied in the context of
232:, which provides a connection between harmonic analysis and
837:"Non-Linear Harmonic Analysis, Operator Theory and P.d.e."
394:
Bass-guitar time signal of open-string A note (55 Hz)
668:
Special functions and the theory of group representation
520:
Non linear harmonic analysis is the use of harmonic and
565:
316:
One of the major results in the theory of functions on
157:, for example as solutions of general, not necessarily
113:
problems, it began to mean waves whose frequencies are
42:
concerned with investigating the connections between a
951:
50:. The frequency representation is found by using the
884:
Introduction to Fourier Analysis on Euclidean Spaces
807:(2nd ed.). New York, NY: Springer. p. 37.
720:
https://www.math.ucla.edu/~tao/247a.1.06f/notes0.pdf
709:
https://www.math.ucla.edu/~tao/247a.1.06f/notes0.pdf
54:
for functions on unbounded domains such as the full
927:, Third edition. Cambridge University Press, 2004.
966:
350:showing a certain understanding of the underlying
609:for computing periodicity in unevenly spaced data
841:Beijing Lectures in Harmonic Analysis. (AM-112)
598:for computing periodicity in evenly-spaced data
169:that may imply their symmetry or periodicity.
117:of one another, as are the frequencies of the
1030:
305:, which can be generalized to a transform of
8:
528:. This includes both problems with infinite
301:. The core motivating ideas are the various
998:M. Bujosa, A. Bujosa and A. Garcıa-Ferrer.
258:Continuous/aperiodic–continuous/aperiodic:
1037:
1023:
1015:
109:, meaning "skilled in music". In physical
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954:
953:
950:
137:. This terminology was extended to other
426:The experimental approach is usually to
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389:
252:Discrete/aperiodic–continuous/periodic:
246:Continuous/periodic–discrete/aperiodic:
28:For broader coverage of this topic, see
629:
141:that solved related equations, then to
1004:IEEE Transactions on Signal Processing
748:A Course in Abstract Harmonic Analysis
338:and second to the case of non-abelian
16:Study of superpositions in mathematics
240:Discrete/periodic–discrete/periodic:
7:
925:An introduction to harmonic analysis
835:Coifman, R. R.; Meyer, Yves (1987).
917:, Princeton University Press, 1970.
907:, Princeton University Press, 1993.
777:"A More Accurate Fourier Transform"
133:first referred to the solutions of
311:locally compact topological groups
14:
356:Non-commutative harmonic analysis
967:{\displaystyle \mathbb {Z} _{2}}
552:
225:in a harmonic-analysis setting.
125:Development of Harmonic Analysis
1169:Least-squares spectral analysis
1096:Fundamental theorem of calculus
607:Least-squares spectral analysis
254:Discrete-time Fourier transform
524:tools and techniques to study
163:partial differential equations
1:
591:Convergence of Fourier series
683:Atiyah-Singer index theorem
642:Online Etymology Dictionary
613:Spectral density estimation
478:hearing the shape of a drum
472:, and (to a lesser extent)
1286:
1006:vol. 63 (2015), 6498–6509.
888:Princeton University Press
680:
266:Abstract harmonic analysis
242:Discrete Fourier transform
176:
46:and its representation in
27:
18:
1234:
1134:
1053:
849:10.1515/9781400882090-002
386:Applied harmonic analysis
903:with Timothy S. Murphy,
119:harmonics of music notes
801:Terras, Audrey (2013).
1101:Calculus of variations
1074:Differential equations
993:Bull. Amer. Math. Soc.
968:
602:Harmonic (mathematics)
403:
395:
321:locally compact groups
278:(for instance via the
195:tempered distributions
101:" originated from the
30:Harmonic (mathematics)
1194:Representation theory
1153:quaternionic analysis
1149:Hypercomplex analysis
1047:mathematical analysis
969:
507:to higher dimensions.
416:differential equation
401:
393:
382:play a crucial role.
354:structure. See also:
309:defined on Hausdorff
284:representation theory
223:uncertainty principle
76:representation theory
1126:Table of derivatives
949:
666:N. Vilenkin (1968).
532:and also non linear
207:Paley–Wiener theorem
1268:Musical terminology
1206:Continuous function
1159:Functional analysis
522:functional analysis
498:spherical harmonics
420:system of equations
288:functional analysis
234:functional analysis
167:boundary conditions
1238:Mathematics portal
1121:Lists of integrals
995:3 (1980), 543–698.
977:Yurii I. Lyubich.
964:
921:Yitzhak Katznelson
746:Gerald B Folland.
564:. You can help by
530:degrees of freedom
404:
396:
363:Plancherel theorem
347:Peter–Weyl theorem
336:topological groups
329:Fourier transforms
325:Pontryagin duality
303:Fourier transforms
299:topological groups
147:elliptic operators
135:Laplace's equation
131:harmonic functions
1263:Harmonic analysis
1245:
1244:
1211:Special functions
1174:Harmonic analysis
943:Fourier Transform
858:978-1-4008-8209-0
843:. pp. 1–46.
582:
581:
526:nonlinear systems
515:Langlands program
511:Automorphic forms
485:Fourier transform
437:Fourier transform
280:Fourier transform
260:Fourier transform
185:Fourier transform
139:special functions
115:integer multiples
84:quantum mechanics
80:signal processing
52:Fourier transform
36:Harmonic analysis
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1164:Fourier analysis
1144:Complex analysis
1045:Major topics in
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374:. In this case,
179:Fourier Analysis
173:Fourier Analysis
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1069:Differentiation
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783:. 2015-07-07
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428:acquire data
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378:in infinite
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215:distribution
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92:neuroscience
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1064:Integration
939:Terence Tao
911:Elias Stein
901:Elias Stein
880:Guido Weiss
876:Elias Stein
820:12 December
781:SourceForge
454:eigenvalues
153:defined on
145:of general
40:mathematics
1252:Categories
1089:stochastic
787:2024-08-26
681:See also:
637:"harmonic"
624:References
380:dimensions
340:Lie groups
323:is called
111:eigenvalue
107:harmonikos
97:The term "
1258:Acoustics
1201:Functions
538:equations
534:operators
470:manifolds
462:Laplacian
441:harmonics
307:functions
276:rotations
155:manifolds
99:harmonics
64:intervals
56:real line
48:frequency
1226:Infinity
1079:ordinary
1059:Calculus
890:, 1971.
585:See also
573:May 2024
295:analysis
159:elliptic
44:function
1084:partial
466:domains
460:of the
412:strings
333:abelian
318:abelian
21:Harmony
1221:Series
931:
894:
855:
811:
474:graphs
205:. The
58:or by
1216:Limit
408:tides
352:group
105:word
929:ISBN
892:ISBN
878:and
853:ISBN
822:2017
809:ISBN
536:and
496:and
456:and
286:and
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568:.
487:on
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217:of
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