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U matematici, Furijeova analiza^[1] je proučavanje načina na koji se opšte funkcije mogu predstaviti ili aproksimirati sumama jednostavnijih trigonometrijskih funkcija. Furijeova analiza je izrasla iz proučavanja Furijeovog reda i nazvana je po Žozefu Furijeu, koji je pokazao da predstavljanje funkcije kao sume trigonometrijskih funkcija uveliko pojednostavljuje proučavanje prenosa toplote.

U današnje vreme, predmet Furijeove analize obuhvata širok matematički spektar. U nauci i inženjerstvu, proces dekompozicije funkcije u oscilatorne komponente se često naziva Furijeova analiza, dok je operacija ponovne izgradnje funkcije iz ovih delova poznata kao Furijeova sinteza. Na primer, određivanje koje su komponente frekvencija prisutne u muzičkoj noti uključivalo bi izračunavanje Furijeove transformacije date muzičke note. Zatim se može resintetisati isti zvuk uključivanjem frekventnih komponenti koje su otkrivene u Furijeovoj analizi. U matematici, termin Furijeova analiza često se odnosi na proučavanje obe operacije.

Proces dekompozicije se naziva Furijeova transformacija.^[2]^[3] Njegov izlaz, Furijeov transformat, često dobija specifičniji naziv, koji zavisi od domena i drugih svojstava funkcije koja se transformiše. Štaviše, originalni koncept Furijeove analize je vremenom proširen kako bi se primenio na sve više apstraktnih i opštih situacija, a generalno polje se često naziva harmonijska analiza. Svaka transformacija koja se koristi za analizu (pogledajte spisak Furijeovih transformacija) ima odgovarajuću inverznu transformaciju koja se može koristiti za sintezu.

Aplikacije

Furijeova analiza ima mnoge naučne primene – u fizici, parcijalnim diferencijalnim jednačinama, teoriji brojeva, kombinatorici, obradi signala, digitalnoj obradi slika, teoriji verovatnoće, statistici, forenzici, vrednovanju deonica, kriptografiji, numeričkoj analizi, akustici, okeanografiji, sonarima, optici, difrakciji, geometriji, analizi proteinske strukture, i drugim oblastima.

Ova široka primenljivost proizilazi iz mnogih korisnih svojstava transformacije:

Transformacije su linearni operatori i uz pravilnu normalizaciju one su i unitarne (svojstvo poznato kao Parsevalova teorema ili, opštenitije kao Planšerelova teorema, i najgeneralnije u vidu Pontrjaginove dualnosti) Rudin 1990.
Transformacije su obično invertibilne.
Eksponencijalne funkcije su svojstvene funkcije diferencijacije, što znači da ova reprezentacija pretvara linearne diferencijalne jednačine sa konstantnim koeficijentima u obične algebarske Evans 1998. Stoga se ponašanje linearnog vremenski invarijantnog sistema može analizirati na svakoj frekvenciji nezavisno.
Prema teoremi konvolucija, Furijeove transformacije pretvaraju komplikovanu operaciju konvolucije u jednostavno množenje, što znači da one pružaju efikasan način za izračunavanje operacija zasnovanih na konvoluciji kao što je polinimijsko množenje i množenje velikih brojeva Knuth 1997.
Diskretna verzija Furijeove transformacije (vidi ispod) može se brzo izvršiti na računarima koristeći algoritme brze Furijeove transkformacije (FFT). Conte & de Boor 1980

U forenzici, laboratorijski infracrveni spektrofotometri koriste analizu Furijeove transformacije za merenje talasnih dužina svetlosti na kojima materijal apsorbuje u infracrvenom spektru. FT metod se koristi za dekodiranje izmerenih signala i zapisivanje podataka o talasnim dužinama. Koristeći kompjuter, ovi Furijeovi proračuni se brzo izvode, tako da za nekoliko sekundi, kompjuterski upravljani FT-IR instrument može da proizvede infracrveni apsorpcioni patern koji je uporediv sa instrumentom sa prizmom.^[4]

Furijeova transformacija je isto tako korisna kao kompaktna reprezentacija signala. Na primer, JPEG kompresija koristi varijantu Furijeove transformacije (diskretna kosinusna transformacija) malih kvadratnih delova digitalne slike. Furijeove komponente svakog kvadrata se zaokružuju na nižu aritmetičku preciznost, a slabe komponente se potpuno eliminišu, tako da se preostale komponente mogu skladištiti veoma kompaktno. U rekonstrukciji slike, svaki kvadrat slike se rekonstruiše iz sačuvanih približnih Furijeovih transformisanih komponenti, koje su inverzno transformišu da bi proizvela aproksimacija originalne slike.^[5]

Reference

^ „Fourier”. Dictionary.com Unabridged. Random House.
^ Khare, Kedar; Butola, Mansi; Rajora, Sunaina (2023). „Chapter 2.3 Fourier Transform as a Limiting Case of Fourier Series”. Fourier Optics and Computational Imaging (2nd изд.). Springer. стр. 13—14. ISBN 978-3-031-18353-9. S2CID 255676773. doi:10.1007/978-3-031-18353-9.
^ Bailey, David H.; Swarztrauber, Paul N. (1994), „A fast method for the numerical evaluation of continuous Fourier and Laplace transforms” (PDF), SIAM Journal on Scientific Computing, 15 (5): 1105—1110, Bibcode:1994SJSC...15.1105B, CiteSeerX 10.1.1.127.1534 , doi:10.1137/0915067, Архивирано из оригинала (PDF) 20. 07. 2008. г., Приступљено 2017-11-01
^ Saferstein, Richard (2013). Criminalistics: An Introduction to Forensic Science.
^ Rabiner, Lawrence R.; Gold, Bernard (1975). Theory and Application of Digital Signal Processing. Englewood Cliffs, NJ. ISBN 9780139141010.

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Spoljašnje veze

Tables of Integral Transforms at EqWorld: The World of Mathematical Equations.
An Intuitive Explanation of Fourier Theory by Steven Lehar.
Lectures on Image Processing: A collection of 18 lectures in pdf format from Vanderbilt University. Lecture 6 is on the 1- and 2-D Fourier Transform. Lectures 7–15 make use of it., by Alan Peters
Moriarty, Philip; Bowley, Roger (2009). „∑ Summation (and Fourier Analysis)”. Sixty Symbols. Brady Haran for the University of Nottingham.

[1] „Fourier”. Dictionary.com Unabridged. Random House.

[:0-2] Khare, Kedar; Butola, Mansi; Rajora, Sunaina (2023). „Chapter 2.3 Fourier Transform as a Limiting Case of Fourier Series”. Fourier Optics and Computational Imaging (2nd изд.). Springer. стр. 13—14. ISBN 978-3-031-18353-9. S2CID 255676773. doi:10.1007/978-3-031-18353-9.

[3] Bailey, David H.; Swarztrauber, Paul N. (1994), „A fast method for the numerical evaluation of continuous Fourier and Laplace transforms” (PDF), SIAM Journal on Scientific Computing, 15 (5): 1105—1110, Bibcode:1994SJSC...15.1105B, CiteSeerX 10.1.1.127.1534 , doi:10.1137/0915067, Архивирано из оригинала (PDF) 20. 07. 2008. г., Приступљено 2017-11-01

[4] Saferstein, Richard (2013). Criminalistics: An Introduction to Forensic Science.

[Rabiner-5] Rabiner, Lawrence R.; Gold, Bernard (1975). Theory and Application of Digital Signal Processing. Englewood Cliffs, NJ. ISBN 9780139141010.

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