Harmonic analysis

For the process of determining the structure of a piece of these requirements in terms of the Fourier transform of

music, see Harmony.

f. The Paley–Wiener theorem is an example of this. The

Harmonic analysis is a branch of mathematics con- Paley–Wiener theorem immediately implies that if f is

a nonzero distribution of compact support (these include

Light Harmonics

functions of compact support), then its Fourier transform

is never compactly supported. This is a very elementary

form of an uncertainty principle in a harmonic analysis

setting. See also: Convergence of Fourier series.

650nm − Red

607nm − Orange

578nm − Yellow

520nm − Green

487nm − Blue

475nm − Indigo

Fourier series can be conveniently studied in the context

of Hilbert spaces, which provides a connection between

harmonic analysis and functional analysis.

405nm − Violet

325nm − UV

Har monics of red light in relation to other colors

λ/14 −

λ/8 −

λ/4 −

1 Applied harmonic analysis

λ/3 −

λ/2.67 −

λ/2.5 −

λ/2 −

White light (all colors interfer ing with each other)

The harmonics of color. The harmonic-analysis chart shows

how the diﬀerent wavelengths interact with red light. At a difference of λ/2 (wavelength/2), red is perfectly in sync with its

second-generation harmonic in the ultraviolet. All other wavelengths in the visual spectrum have less than a λ/2 diﬀerence

between them, forming harmonic oscillations in the combined

waves. At λ/14, the oscillations will cycle every fourteenth wave,

while at λ/8 they will cycle every eighth. The oscillations are

most rapid at λ/4, cycling every fourth wave, while at λ/3 they

cycle every seventh wave, and at λ/2.5 they cycle every thirteenth.

The lower section shows how the λ/4 harmonic interacts in visible

light (green and red), as photographed in an optical ﬂat.

Bass guitar time signal of open string A note (55 Hz).

cerned with the representation of functions or signals as

the superposition of basic waves, and the study of and

generalization of the notions of Fourier series and Fourier

transforms (i.e. an extended form of Fourier analysis).

In the past two centuries, it has become a vast subject

with applications in areas as diverse as signal processing,

quantum mechanics, tidal analysis and neuroscience.

The term "harmonics" originated as the ancient Greek

word, “harmonikos,” meaning “skilled in music.”[1] In

physical eigenvalue problems it began to mean waves

whose frequencies are integer multiples of one another, as

are the frequencies of the harmonics of music notes, but

the term has been generalized beyond its original meaning.

Fourier transform of bass guitar time signal of open string A

note (55 Hz), computed with https://sourceforge.net/projects/

amoreaccuratefouriertransform/ .

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 tides and vibrating strings

are common and simple examples. 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

The classical Fourier transform on Rn is still an area of

ongoing research, particularly concerning Fourier transformation on more general objects such as tempered distributions. For instance, if we impose some requirements on a distribution f, we can attempt to translate

1

2

5

REFERENCES

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.

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,

The experimental approach is usually to acquire data for example SLn. In this case, representations in inﬁnite

that accurately quantiﬁes the phenomenon. For exam- dimensions play a crucial role.

ple, 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 3 Other branches

a long enough duration that multiple oscillatory periods

are likely included. In a study on vibrating strings, it is

• Study of the eigenvalues and eigenvectors of the

common for the experimentalist to acquire a sound waveLaplacian on domains, manifolds, and (to a lesser

form sampled at a rate at least twice that of the highest

extent) graphs is also considered a branch of harfrequency expected and for a duration many times the pemonic analysis. See e.g., hearing the shape of a

riod of the lowest frequency expected.

drum.

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 fundamental frequency or 55 Hz. 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, which

is shown in the lower ﬁgure. Note that there is a prominent peak at 55 Hz, but that there are 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 harmonics.

2

Abstract harmonic analysis

One of the most modern branches of harmonic analysis,

having its roots in the mid-twentieth 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 nonabelian 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. See also:

Non-commutative harmonic analysis.

• Harmonic analysis on Euclidean spaces deals with

properties of the Fourier transform on Rn 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 Bessel

functions and spherical harmonics.

• Harmonic analysis on tube domains is concerned

with generalizing properties of Hardy spaces to

higher dimensions.

4 See also

• Harmonic (mathematics)

• Spectral density estimation

5 References

[1] http://www.etymonline.com/index.php?term=harmonic

• Elias Stein and Guido Weiss, Introduction to Fourier

Analysis on Euclidean Spaces, Princeton University

Press, 1971. ISBN 0-691-08078-X

• Elias Stein with Timothy S. Murphy, Harmonic

Analysis: Real-Variable Methods, Orthogonality,

and Oscillatory Integrals, Princeton University

Press, 1993.

• Elias Stein, Topics in Harmonic Analysis Related to

the Littlewood-Paley Theory, Princeton University

Press, 1970.

• Yitzhak Katznelson, An introduction to harmonic

analysis, Third edition. Cambridge University

Press, 2004. ISBN 0-521-83829-0; 0-521-543592

3

• Terence Tao, Fourier Transform. (Introduces the

decomposition of functions into odd + even parts as

a harmonic decomposition over ℤ₂.)

• Yurii I. Lyubich. Introduction to the Theory of

Banach Representations of Groups. Translated

from the 1985 Russian-language edition (Kharkov,

Ukraine). Birkhäuser Verlag. 1988.

4

6 TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

6

Text and image sources, contributors, and licenses

6.1

Text

• Harmonic analysis Source: https://en.wikipedia.org/wiki/Harmonic_analysis?oldid=702632299 Contributors: AxelBoldt, Zundark,

ChangChienFu, Michael Hardy, Gabbe, Looxix~enwiki, Stevan White, Stevenj, Rob Hooft, Revolver, Charles Matthews, Dysprosia,

Hyacinth, Samsara, Robbot, Sverdrup, Giftlite, BenFrantzDale, CSTAR, D6, Gadykozma, Gauge, DanP, Bobo192, La goutte de pluie,

Drf5n, Msh210, Linas, Guardian of Light, BD2412, FlaBot, Jeﬀ02, Krishnavedala, Sunev, YurikBot, RobotE, Gaius Cornelius, Wiki

alf, Tong~enwiki, Dhollm, Crasshopper, Levnir, Kompik, Reyk, Banus, SmackBot, Saihtam, Chrislewis.au, Chlewbot, Mhym, Radagast83, LaMenta3, Zero sharp, Prof.Maque, Basawala, Tawkerbot4, Thijs!bot, Thenub314, VoABot II, David Eppstein, R'n'B, Maurice

Carbonaro, Vanished user 39948282, JohnBlackburne, Random3f, Digby Tantrum, Seraphita~enwiki, Addbot, Fgnievinski, Topology Expert, CarsracBot, Zaereth, Omnipaedista, Kiefer.Wolfowitz, EmausBot, KHamsun, Slawekb, Mmqte, Boashash, MerlIwBot, Solomon7968,

Brad7777, Declaration1776, Lemnaminor, Erru il 1988, Sheliazhenko, Niki Goss, KasparBot, Fourier1789, Headabot and Anonymous:

51

6.2

Images

• File:Bass_Guitar_Time_Signal_of_open_string_A_note_(55_Hz).png Source: https://upload.wikimedia.org/wikipedia/commons/a/

a6/Bass_Guitar_Time_Signal_of_open_string_A_note_%2855_Hz%29.png License: CC BY-SA 4.0 Contributors: Own work Original

artist: Fourier1789

• File:Fourier_Transform_of_bass_guitar_time_signal.png Source: https://upload.wikimedia.org/wikipedia/commons/0/08/Fourier_

Transform_of_bass_guitar_time_signal.png License: CC BY-SA 4.0 Contributors: Own work Original artist: Fourier1789

• File:Light_wave_harmonic_diagram.svg Source:

https://upload.wikimedia.org/wikipedia/commons/7/71/Light_wave_harmonic_

diagram.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Rubber Duck (☮ • ✍)

6.3

Content license

• Creative Commons Attribution-Share Alike 3.0

For the process of determining the structure of a piece of these requirements in terms of the Fourier transform of

music, see Harmony.

f. The Paley–Wiener theorem is an example of this. The

Harmonic analysis is a branch of mathematics con- Paley–Wiener theorem immediately implies that if f is

a nonzero distribution of compact support (these include

Light Harmonics

functions of compact support), then its Fourier transform

is never compactly supported. This is a very elementary

form of an uncertainty principle in a harmonic analysis

setting. See also: Convergence of Fourier series.

650nm − Red

607nm − Orange

578nm − Yellow

520nm − Green

487nm − Blue

475nm − Indigo

Fourier series can be conveniently studied in the context

of Hilbert spaces, which provides a connection between

harmonic analysis and functional analysis.

405nm − Violet

325nm − UV

Har monics of red light in relation to other colors

λ/14 −

λ/8 −

λ/4 −

1 Applied harmonic analysis

λ/3 −

λ/2.67 −

λ/2.5 −

λ/2 −

White light (all colors interfer ing with each other)

The harmonics of color. The harmonic-analysis chart shows

how the diﬀerent wavelengths interact with red light. At a difference of λ/2 (wavelength/2), red is perfectly in sync with its

second-generation harmonic in the ultraviolet. All other wavelengths in the visual spectrum have less than a λ/2 diﬀerence

between them, forming harmonic oscillations in the combined

waves. At λ/14, the oscillations will cycle every fourteenth wave,

while at λ/8 they will cycle every eighth. The oscillations are

most rapid at λ/4, cycling every fourth wave, while at λ/3 they

cycle every seventh wave, and at λ/2.5 they cycle every thirteenth.

The lower section shows how the λ/4 harmonic interacts in visible

light (green and red), as photographed in an optical ﬂat.

Bass guitar time signal of open string A note (55 Hz).

cerned with the representation of functions or signals as

the superposition of basic waves, and the study of and

generalization of the notions of Fourier series and Fourier

transforms (i.e. an extended form of Fourier analysis).

In the past two centuries, it has become a vast subject

with applications in areas as diverse as signal processing,

quantum mechanics, tidal analysis and neuroscience.

The term "harmonics" originated as the ancient Greek

word, “harmonikos,” meaning “skilled in music.”[1] In

physical eigenvalue problems it began to mean waves

whose frequencies are integer multiples of one another, as

are the frequencies of the harmonics of music notes, but

the term has been generalized beyond its original meaning.

Fourier transform of bass guitar time signal of open string A

note (55 Hz), computed with https://sourceforge.net/projects/

amoreaccuratefouriertransform/ .

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 tides and vibrating strings

are common and simple examples. 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

The classical Fourier transform on Rn is still an area of

ongoing research, particularly concerning Fourier transformation on more general objects such as tempered distributions. For instance, if we impose some requirements on a distribution f, we can attempt to translate

1

2

5

REFERENCES

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.

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,

The experimental approach is usually to acquire data for example SLn. In this case, representations in inﬁnite

that accurately quantiﬁes the phenomenon. For exam- dimensions play a crucial role.

ple, 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 3 Other branches

a long enough duration that multiple oscillatory periods

are likely included. In a study on vibrating strings, it is

• Study of the eigenvalues and eigenvectors of the

common for the experimentalist to acquire a sound waveLaplacian on domains, manifolds, and (to a lesser

form sampled at a rate at least twice that of the highest

extent) graphs is also considered a branch of harfrequency expected and for a duration many times the pemonic analysis. See e.g., hearing the shape of a

riod of the lowest frequency expected.

drum.

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 fundamental frequency or 55 Hz. 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, which

is shown in the lower ﬁgure. Note that there is a prominent peak at 55 Hz, but that there are 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 harmonics.

2

Abstract harmonic analysis

One of the most modern branches of harmonic analysis,

having its roots in the mid-twentieth 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 nonabelian 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. See also:

Non-commutative harmonic analysis.

• Harmonic analysis on Euclidean spaces deals with

properties of the Fourier transform on Rn 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 Bessel

functions and spherical harmonics.

• Harmonic analysis on tube domains is concerned

with generalizing properties of Hardy spaces to

higher dimensions.

4 See also

• Harmonic (mathematics)

• Spectral density estimation

5 References

[1] http://www.etymonline.com/index.php?term=harmonic

• Elias Stein and Guido Weiss, Introduction to Fourier

Analysis on Euclidean Spaces, Princeton University

Press, 1971. ISBN 0-691-08078-X

• Elias Stein with Timothy S. Murphy, Harmonic

Analysis: Real-Variable Methods, Orthogonality,

and Oscillatory Integrals, Princeton University

Press, 1993.

• Elias Stein, Topics in Harmonic Analysis Related to

the Littlewood-Paley Theory, Princeton University

Press, 1970.

• Yitzhak Katznelson, An introduction to harmonic

analysis, Third edition. Cambridge University

Press, 2004. ISBN 0-521-83829-0; 0-521-543592

3

• Terence Tao, Fourier Transform. (Introduces the

decomposition of functions into odd + even parts as

a harmonic decomposition over ℤ₂.)

• Yurii I. Lyubich. Introduction to the Theory of

Banach Representations of Groups. Translated

from the 1985 Russian-language edition (Kharkov,

Ukraine). Birkhäuser Verlag. 1988.

4

6 TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

6

Text and image sources, contributors, and licenses

6.1

Text

• Harmonic analysis Source: https://en.wikipedia.org/wiki/Harmonic_analysis?oldid=702632299 Contributors: AxelBoldt, Zundark,

ChangChienFu, Michael Hardy, Gabbe, Looxix~enwiki, Stevan White, Stevenj, Rob Hooft, Revolver, Charles Matthews, Dysprosia,

Hyacinth, Samsara, Robbot, Sverdrup, Giftlite, BenFrantzDale, CSTAR, D6, Gadykozma, Gauge, DanP, Bobo192, La goutte de pluie,

Drf5n, Msh210, Linas, Guardian of Light, BD2412, FlaBot, Jeﬀ02, Krishnavedala, Sunev, YurikBot, RobotE, Gaius Cornelius, Wiki

alf, Tong~enwiki, Dhollm, Crasshopper, Levnir, Kompik, Reyk, Banus, SmackBot, Saihtam, Chrislewis.au, Chlewbot, Mhym, Radagast83, LaMenta3, Zero sharp, Prof.Maque, Basawala, Tawkerbot4, Thijs!bot, Thenub314, VoABot II, David Eppstein, R'n'B, Maurice

Carbonaro, Vanished user 39948282, JohnBlackburne, Random3f, Digby Tantrum, Seraphita~enwiki, Addbot, Fgnievinski, Topology Expert, CarsracBot, Zaereth, Omnipaedista, Kiefer.Wolfowitz, EmausBot, KHamsun, Slawekb, Mmqte, Boashash, MerlIwBot, Solomon7968,

Brad7777, Declaration1776, Lemnaminor, Erru il 1988, Sheliazhenko, Niki Goss, KasparBot, Fourier1789, Headabot and Anonymous:

51

6.2

Images

• File:Bass_Guitar_Time_Signal_of_open_string_A_note_(55_Hz).png Source: https://upload.wikimedia.org/wikipedia/commons/a/

a6/Bass_Guitar_Time_Signal_of_open_string_A_note_%2855_Hz%29.png License: CC BY-SA 4.0 Contributors: Own work Original

artist: Fourier1789

• File:Fourier_Transform_of_bass_guitar_time_signal.png Source: https://upload.wikimedia.org/wikipedia/commons/0/08/Fourier_

Transform_of_bass_guitar_time_signal.png License: CC BY-SA 4.0 Contributors: Own work Original artist: Fourier1789

• File:Light_wave_harmonic_diagram.svg Source:

https://upload.wikimedia.org/wikipedia/commons/7/71/Light_wave_harmonic_

diagram.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Rubber Duck (☮ • ✍)

6.3

Content license

• Creative Commons Attribution-Share Alike 3.0