Harmonic analysis and applications.
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| Title: | Harmonic analysis and applications. |
|---|---|
| Authors: | Filali, Mahmoud |
| Source: | Kybernetes. 2012, Vol. 41 Issue 1/2, p129-144. 16p. |
| Subjects: | Mathematics, Eigenvalues, Eigenfunctions, Fourier transforms, Harmonic analysis (Mathematics), Trigonometry |
| Abstract: | Purpose – The purpose of this paper is to survey briefly how harmonic analyis started and developed throughout the centuries to reach its modern status and its surprisingly wide range of applications. Design/methodology/approach – The author traces applications of harmonic analysis back to Mesopotamia, ancient Egypt and the Indus Valley, showing how the Greeks have applied trigonometry and influenced its birth, then the important developments in India in the sixth century laying the first brick to modern trigonometry with the definition of the sinus, then medieval India founding modern mathematical analysis. Trigonometry was developed further by the Arabs until the fourteenth century, then by the Europeans. The eighteenth century in France was particularly important when Bernoulli solved, with an infinite trigonometric series, the vibrating string problem, then Fourier, who studied these series extensively. The author goes on to harmonic analysis on locally compact groups, and ends up with a quick personal view on harmonic analysis nowadays. The last section of the paper presents some of the modern applications. Harmonic analysis is, of course, still used for navigation but also has many other very surprising applications such as signal processing, quantum mechanics, neuroscience, tomography, etc. Findings – The power of harmonic analysis lies in giving the solutions to various problems as infinite series of basic functions, so to be able to produce algorithms for FFT boxes, it must be understood how these series came about and the convergence of these series. Originality/value – The review should be useful to people interested in studying and/or applying harmonic analysis. [ABSTRACT FROM AUTHOR] |
| Copyright of Kybernetes is the property of Emerald Publishing Limited and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
| FullText | Text: Availability: 0 |
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| Header | DbId: egs DbLabel: Engineering Source An: 84775499 AccessLevel: 6 PubType: Periodical PubTypeId: serialPeriodical PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Harmonic analysis and applications. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Filali%2C+Mahmoud%22">Filali, Mahmoud</searchLink> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Kybernetes%22">Kybernetes</searchLink>. 2012, Vol. 41 Issue 1/2, p129-144. 16p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Mathematics%22">Mathematics</searchLink><br /><searchLink fieldCode="DE" term="%22Eigenvalues%22">Eigenvalues</searchLink><br /><searchLink fieldCode="DE" term="%22Eigenfunctions%22">Eigenfunctions</searchLink><br /><searchLink fieldCode="DE" term="%22Fourier+transforms%22">Fourier transforms</searchLink><br /><searchLink fieldCode="DE" term="%22Harmonic+analysis+%28Mathematics%29%22">Harmonic analysis (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Trigonometry%22">Trigonometry</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: Purpose – The purpose of this paper is to survey briefly how harmonic analyis started and developed throughout the centuries to reach its modern status and its surprisingly wide range of applications. Design/methodology/approach – The author traces applications of harmonic analysis back to Mesopotamia, ancient Egypt and the Indus Valley, showing how the Greeks have applied trigonometry and influenced its birth, then the important developments in India in the sixth century laying the first brick to modern trigonometry with the definition of the sinus, then medieval India founding modern mathematical analysis. Trigonometry was developed further by the Arabs until the fourteenth century, then by the Europeans. The eighteenth century in France was particularly important when Bernoulli solved, with an infinite trigonometric series, the vibrating string problem, then Fourier, who studied these series extensively. The author goes on to harmonic analysis on locally compact groups, and ends up with a quick personal view on harmonic analysis nowadays. The last section of the paper presents some of the modern applications. Harmonic analysis is, of course, still used for navigation but also has many other very surprising applications such as signal processing, quantum mechanics, neuroscience, tomography, etc. Findings – The power of harmonic analysis lies in giving the solutions to various problems as infinite series of basic functions, so to be able to produce algorithms for FFT boxes, it must be understood how these series came about and the convergence of these series. Originality/value – The review should be useful to people interested in studying and/or applying harmonic analysis. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Kybernetes is the property of Emerald Publishing Limited and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1108/03684921211213160 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 16 StartPage: 129 Subjects: – SubjectFull: Mathematics Type: general – SubjectFull: Eigenvalues Type: general – SubjectFull: Eigenfunctions Type: general – SubjectFull: Fourier transforms Type: general – SubjectFull: Harmonic analysis (Mathematics) Type: general – SubjectFull: Trigonometry Type: general Titles: – TitleFull: Harmonic analysis and applications. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Filali, Mahmoud IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 02 Text: 2012 Type: published Y: 2012 Identifiers: – Type: issn-print Value: 0368492X Numbering: – Type: volume Value: 41 – Type: issue Value: 1/2 Titles: – TitleFull: Kybernetes Type: main |
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