Fourierova ?ada slou?i k aproximaci periodicke funkce ?adou harmonickych funkci sinus a kosinus . Zakladni my?lenka zapisu funkce ve form? uvedene ?ady spo?iva v tzv. ortogonalnim rozkladu funkce v linearnim prostoru funkci po ?astech spojitych na intervalu ${\displaystyle \langle 0,T\rangle }$ spolu s definovanym skalarnim sou?inem :

${\displaystyle f\cdot g=\int _{0}^{T}f(t)\ g(t)\ dt}$,

tvo?icich tzv. Hilbert?v prostor , kde ${\displaystyle T}$ je doba periody pr?b?hu funkce.

Fourierova ?ada je pojmenovana po francouzskem fyzikovi a matematikovi Josephu Fourierovi .

Ortogonalni rozklad funkce [ editovat | editovat zdroj ]

M?jme linearni podprostor ${\displaystyle \rho }$ dimenze ${\displaystyle n}$ Hilbertova prostoru nekone?ne dimenze o ortonormalni bazi ${\displaystyle E}$:

${\displaystyle \rho \subset H\,\,\,\,\,\,\,\,\,\,}$${\displaystyle dimH=\infty \,\,\,\,\,\,\,\,\,\,}$${\displaystyle \dim \rho =n\,\,\,\,\,\,\,\,\,\,}$${\displaystyle f\in H\,\,\,\,\,\,\,\,\,\,}$${\displaystyle f_{n}\in \rho \,\,\,\,\,\,\,\,\,\,}$${\displaystyle n=1,2,3,\cdots \,\,\,\,\,\,\,\,\,\,}$${\displaystyle E=e_{1},e_{2},e_{3},\cdots }$

pak pro Euklidovskou vzdalenost funkci ${\displaystyle f}$ a ${\displaystyle f_{n}}$ plati:

${\displaystyle \left|f-f_{n}\right|^{2}=\left(f-f_{n}\right)\cdot \left(f-f_{n}\right)=f\cdot f-2f\cdot f_{n}+f_{n}\cdot f_{n}=f\cdot f-2f\cdot \sum _{i}^{}{s_{i}e_{i}}+\sum _{i}^{}s_{i}^{2}=}$

${\displaystyle =f\cdot f+\sum _{i}^{}\left({f\cdot e}_{i}\right)^{2}-2\sum _{i}^{}{s_{i}\left(f\cdot e_{i}\right)}+\sum _{i}^{}s_{i}^{2}-\sum _{i}^{}\left({f\cdot e}_{i}\right)^{2}=f\cdot f+\sum _{i}^{}\left(\left(f\cdot e_{i}\right)-s_{i}\right)^{2}-\sum _{i}^{}\left({f\cdot e}_{i}\right)^{2}\,\,\,\,\,\,\,\,\,\,}$ kde ${\displaystyle i=1,\cdots ,n}$

a

${\displaystyle s_{i}=f\cdot e_{i}\Rightarrow \left|f-f_{n}\right|^{2}=f\cdot f-\sum _{i}^{}\left({f\cdot e}_{i}\right)^{2}=\left|f\right|^{2}-\left|f_{n}\right|^{2}\Rightarrow \left|f\right|^{2}\geq \left|f_{n}\right|^{2}\Rightarrow f\sim f_{n}}$

kde ${\displaystyle s_{1},\cdots ,s_{n}}$ jsou sou?adnice ${\displaystyle f_{n}}$ vzhledem k ${\displaystyle E}$, pak m??eme aproximovat funkci ${\displaystyle f}$ nasledujici ?adou:

${\displaystyle \lim _{n\rightarrow \infty }\left|f-f_{n}\right|^{2}=0\Rightarrow \left|f\right|^{2}\approx \left|f_{n}\right|^{2}\Rightarrow f\approx f_{n}=\sum _{i}^{}{\left(f\cdot e_{i}\right)\ e_{i}}\,\,\,\,\,\,\,\,\,\,}$ kde ${\displaystyle i=1,\cdots ,n}$

Fourierova ?ada v goniometrickem tvaru [ editovat | editovat zdroj ]

Mno?ina ${\displaystyle \left\{{\frac {1}{\sqrt {T}}},{{\frac {\sqrt {2}}{\sqrt {T}}}\cos }\omega t,{{\frac {\sqrt {2}}{\sqrt {T}}}\sin }\omega t,{{\frac {\sqrt {2}}{\sqrt {T}}}\cos }{2\omega t},{{\frac {\sqrt {2}}{\sqrt {T}}}\sin }{2\omega t},{{\frac {\sqrt {2}}{\sqrt {T}}}\cos }{3\omega t},{{\frac {\sqrt {2}}{\sqrt {T}}}\sin }{3\omega t},\cdots \right\}}$ tvo?i ortonormalni bazi vy?e uvedeneho Hilbertova prostoru nekone?ne dimenze, pak funkci ${\displaystyle f}$ m??eme aproximovat pomoci nasledujici goniometricke ?ady:

${\displaystyle f\left(t\right)\approx \sum _{0}^{\infty }{a_{n}\cos n\omega t+b_{n}\sin n\omega t}\,\,\,\,\,\,\,\,\,\,}$${\displaystyle n\mathbb {\in N} }$

kde ${\displaystyle a_{0}={\frac {1}{T}}\int _{0}^{T}{f\left(t\right)\,dt}\,\,\,\,\,\,\,\,\,\,}$ ${\displaystyle a_{n}={\frac {2}{T}}\int _{0}^{T}{f\left(t\right)\cos n\omega t\,\,dt}\,\,\,\,\,\,\,\,\,\,}$ ${\displaystyle b_{n}={\frac {2}{T}}\int _{0}^{T}{f\left(t\right)\sin n\omega t\,\,dt}\,\,\,\,\,\,\,\,\,\,}$ pro ${\displaystyle n=1,2,3,\cdots }$.

Koeficient ${\displaystyle b_{0}}$ nema smysl uva?ovat, nebo? ${\displaystyle b_{0}\sin 0=0}$.

Pokud se dv? integrovatelne funkce li?i v kone?nem po?tu bod?, tak je jasne, ?e maji stejnou Fourierovu ?adu. Z toho d?vodu nepi?eme mezi funkci ${\displaystyle f}$ a jeji Fourierovou ?adou rovnitko. Pokud je v?ak funkce vybrana z obecn?j?i mno?iny ne? jen z mno?iny integrovatelnych funkci, tak se ji Fourierova ?ada m??e rovnat. Nap?iklad plati nasledujici tvrzeni: pokud je funkce ${\displaystyle f}$ ohrani?ena a po ?astech spojita a ma i ohrani?enou po ?astech spojitou prvni derivaci , tak jeji Fourierova ?ada ma v ka?dem bod? sou?et, a ten je roven aritmetickemu pr?m?ru prave a leve limity teto funkce v tomto bod?. Tedy v bod? spojitosti je to hodnota funkce. Fourierova ?ada spojite funkce nemusi (v n?kterem bod?) v?bec konvergovat.

V praxi se funkce ${\displaystyle f}$ aproximuje kone?nym rozvojem, kde s?itame jen n?kolik prvnich ?len?, ?im? se genericky s nar?stajicim po?tem ?len? zvy?uje p?esnost teto aproximace.

P?iklad [ editovat | editovat zdroj ]

M?jme exponencialu zu?en? definovanou na intervalu ${\displaystyle <0,1>}$ a vytvo?me z ni sudou a lichou periodickou funkci s periodou ${\displaystyle T=2}$ na intervalu ${\displaystyle <-1,1>}$ a uhlovou frekvenci ${\displaystyle \omega =\pi }$, pak m??eme uvedenou sudou a lichou funkci aproximovat nasledujicimi ?adami:

suda funkce :

${\displaystyle a_{0}={\frac {1}{2}}(\int _{-1}^{0}{e^{-t}\,dt}+\int _{0}^{1}{e^{t}\,dt})=e-1}$

${\displaystyle a_{n}=\int _{-1}^{0}{e^{-t}\cos n\pi t\,\,dt}+\int _{0}^{1}{e^{t}\cos n\pi t\,\,dt}=2{\frac {(-1)^{n}e-1}{(n\pi )^{2}+1}}}$

${\displaystyle b_{n}=0}$

kde ${\displaystyle n=1,2,3,\cdots }$

${\displaystyle f\left(t\right)\approx (e-1)+2\sum _{n=1}^{\infty }{{\frac {(-1)^{n}e-1}{(n\pi )^{2}+1}}\cos n\pi t}}$

licha funkce :

${\displaystyle a_{0}=0}$

${\displaystyle a_{n}=0}$

${\displaystyle b_{n}=-\int _{-1}^{0}{e^{-t}\sin n\pi t\,\,dt}+\int _{0}^{1}{e^{t}\sin n\pi t\,\,dt}=-2n\pi {\frac {(-1)^{n}e-1}{(n\pi )^{2}+1}}}$

kde ${\displaystyle n=1,2,3,\cdots }$

${\displaystyle f\left(t\right)\approx -2\sum _{n=1}^{\infty }{n\pi {\frac {(-1)^{n}e-1}{(n\pi )^{2}+1}}\sin n\pi t}}$

Poznamenejme, ?e Fourierova ?ada sude resp. liche funkce obsahuje pouze ?leny s funkci cosinus resp. sinus.

Fourierova ?ada v exponencialnim tvaru [ editovat | editovat zdroj ]

Z nasledujicich vztah?:

${\displaystyle e^{in\omega t}+e^{-in\omega t}=\left(\cos n\omega t+i\sin n\omega t\right)+\left(\cos n\omega t-i\sin n\omega t\right)=2\cos n\omega t}$

${\displaystyle e^{in\omega t}-e^{-in\omega t}=\left(\cos n\omega t+i\sin n\omega t\right)-\left(\cos n\omega t-i\sin n\omega t\right)=i2\sin n\omega t}$

a

${\displaystyle a_{n}\cos n\omega t+b_{n}\sin n\omega t={\frac {a_{n}}{2}}\left(e^{in\omega t}+e^{-in\omega t}\right)-i{\frac {b_{n}}{2}}\left(e^{in\omega t}-e^{-in\omega t}\right)=}$

${\displaystyle ={\frac {1}{2}}\left(a_{n}-ib_{n}\right)e^{in\omega t}+{\frac {1}{2}}\left(a_{n}+ib_{n}\right)e^{-in\omega t}=\left({c_{n}e}^{in\omega t}+{\overline {c}}_{n}e^{-in\omega t}\right)}$

dostaneme:

${\displaystyle 2c_{n}=\left(a_{n}-ib_{n}\right)={\frac {2}{T}}\int _{0}^{T}{f\left(t\right)\left(\cos n\omega t-i\sin n\omega t\right)dt}\Rightarrow c_{n}={\frac {1}{T}}\int _{0}^{T}{f\left(t\right)\ e^{-in\omega t}dt}}$

${\displaystyle 2{\overline {c}}_{n}=\left(a_{n}+ib_{n}\right)={\frac {2}{T}}\int _{0}^{T}{f\left(t\right)\left(\cos n\omega t+i\sin n\omega t\right)dt}\Rightarrow {\overline {c}}_{n}={\frac {1}{T}}\int _{0}^{T}{f\left(t\right)\ e^{in\omega t}dt}}$,

tak?e potom m??eme vyjad?it aproximaci funkce ${\displaystyle f}$ pomoci nasledujici exponencialni ?ady:

${\displaystyle f\left(t\right)\approx \sum _{-\infty }^{\infty }c_{n}e^{in\omega t}\,\,\,\,\,\,\,\,\,\,}$${\displaystyle n\mathbb {\in Z} \,\,\,\,\,\,\,\,\,\,}$ kde ${\displaystyle a_{0}=c_{0}={\overline {f}}}$ je st?edni hodnota funkce ${\displaystyle f}$.

Parsevalova rovnost [ editovat | editovat zdroj ]

Nech?

${\displaystyle f\left(t\right)\approx \sum _{0}^{\infty }{a_{n}\cos n\omega t+b_{n}\sin n\omega t}=\sum _{-\infty }^{\infty }c_{n}e^{in\omega t}\,\,\,\,\,\,\,\,\,\,}$${\displaystyle n\mathbb {\in Z} }$.

Pak plati nasledujici Parsevalova rovnost , vyjad?ujici, ?e efektivni hodnota aproximovane funkce (st?edni hodnota jejiho ?tverce) je rovna sum? kvadrat? koeficient? aproximujici Fourierovy ?ady:

${\displaystyle {\frac {1}{T}}\int _{0}^{T}f^{2}(t)\,\mathrm {d} t=\sum _{n=0}^{\infty }\left(a_{n}^{2}+b_{n}^{2}\right)=\sum _{n=-\infty }^{\infty }c_{n}^{2}}$.

Jmeno tomuto tvrzeni dal francouzsky matematik Marc-Antoine Parseval . Pokud levou stranu rovnice interpretujeme jako ?tverec normy funkce ${\displaystyle f}$, lze Parsevalovu rovnost ?ist jako zobecn?ni Pythagorovy v?ty na nekone?n?rozm?rny prostor funkci.

Fourierova transformace [ editovat | editovat zdroj ]

Ze vztahu doby periody bli?ici se nekone?nu a uhlove frekvence sit?:

${\displaystyle T\omega =2\pi \Rightarrow \left(T\rightarrow \infty \Rightarrow \omega \rightarrow 0\right)\Rightarrow n\omega \equiv \omega \equiv d\omega }$

lze zavest u?itim limitnich p?echod? spojitou Fourierovu transformaci:

${\displaystyle \lim _{T\rightarrow \infty }{Tc_{n}}=\lim _{T\rightarrow \infty }{\int _{-{\frac {T}{2}}}^{\frac {T}{2}}{f\left(t\right)e^{-in\omega t}dt}}=\int _{-\infty }^{\infty }{f\left(t\right)\ e^{-i\omega t}dt}=F\left(\omega \right)}$

a naopak inverzni spojitou Fourierovu transformaci:

${\displaystyle f\left(t\right)=\lim _{T\rightarrow \infty }{\sum _{-\infty }^{\infty }{T{\frac {\omega }{2\pi }}c_{n}e^{in\omega t}}}={\frac {1}{2\pi }}\sum _{-\infty }^{\infty }{\lim _{T\rightarrow \infty }{Tc_{n}e^{in\omega t}\omega }}={\frac {1}{2\pi }}\int _{-\infty }^{\infty }{F\left(\omega \right)\ e^{in\omega t}{\text{dω}}}}$

Odkazy [ editovat | editovat zdroj ]

Literatura [ editovat | editovat zdroj ]

BARTSCH, Hans-Jochen. Matematicke vzorce . 4. vyd. Praha: Academia, 1994. 832 s. ISBN   80-200-1448-9 .  

Souvisejici ?lanky [ editovat | editovat zdroj ]

• Fourierova transformace
• Rychla Fourierova transformace

Externi odkazy [ editovat | editovat zdroj ]

• Obrazky, zvuky ?i videa k tematu Fourierova ?ada na Wikimedia Commons
• Fourier Series 3D - interaktivni demonstrace principu Fourierovych ?ad HTML5 a JavaScript: Unikatni interaktivni 3D zobrazeni propojujici ?asovou, frekven?ni, amplitudovou a fazovou osu.