Dérét Fourier: Béda antarrépisi

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Baris ka-85:
==Vérsi modéren nu ngagunakeun éksponénsial kompléks ===
 
Ku cara ngagunakeun [[rumus Euler]], <math>e^{intjnt}=\cos(nt)+ij\sin(nt)</math>, dimana <math>i</math> nyaéta [[unit imajinér]], urang bisa ngagambarkeun dérét Fourier jadi rumus nu leuwih ringkes''':'''
 
Upamana waé cx(t) ngarupakeun hiji sinyal périodik kalayan périoda T<sub>o</sub>. Mangka urang ngadéfinisikeun dérét Fourier éksponénsial kompléks x(t) minangka
 
:<math>x(t) = \sum_{n=-\infty}^{\infty} c_n e^{int}.</math>
Baris ka-103:
:<math>b_n = i( c_{n} - c_{-n} )</math> pikeun <math>n=1,2,\dots</math>
 
The notation <math>c_n</math> is inadequate for discussing the Fourier coefficients of several different functions. Therefore it is customarily replaced by a modified form of <math>f\,</math> (in this case), such as <math>F\,</math> or <math>\hat{f},</math>&nbsp; and functional notation often replaces subscripting.&nbsp; Thus''':'''
 
:<math>
\begin{align}
f(x) &= \sum_{n=-\infty}^{\infty} \hat{f}(n)\cdot e^{inx} \\
&= \sum_{n=-\infty}^{\infty} F[n]\cdot e^{inx} \quad \mbox{(engineering)}.
\end{align}
</math>
 
In various fields of science, the sequence has other names, such as [[characteristic function (probability theory)|characteristic function]] (probability theory). In engineering, particularly when variable '''x''' represents time, the sequence is called a [[frequency domain]] representation. Square brackets are often used to emphasize that the domain of this function is a '''discrete''' set of frequencies.
 
=== Fourier series on a general interval [''a'',''b''] ===
 
Let G[0], G[±1], G[±2], '''…''' be real or complex coefficients. The Fourier series''':'''
 
:<math>g(x)=\sum_{n=-\infty}^\infty G[n]\cdot e^{i 2\pi \frac{n}{\tau} x}\,</math>
 
is a periodic function, whose period is <math>\tau\,</math> on the domain <math>\mathbb{R}.</math>&nbsp; If a function is [[square-integrable]] in the interval <math>[a,\ a+\tau],</math>&nbsp; it can be represented in that interval by the formula above. If ''g''(''x'') is integrable, then the Fourier coefficients are given by''':'''
 
:<math>G[n] = \frac{1}{\tau}\int_a^{a+\tau} g(x)\cdot e^{-i 2\pi \frac{n}{\tau} x}\, dx.</math>
 
Note that if the function to be represented is also <math>\tau\,</math>-periodic, then <math>a\,</math> is an arbitrary choice. Two popular choices are <math>a=0\,</math> and <math>a=-\tau/2.\,</math>&nbsp;
 
Another commonly used frequency domain representation uses the Fourier series coefficients to modulate a [[Dirac comb]]''':'''
 
:<math>
G(f) \ \stackrel{\mathrm{def}}{=} \ \sum_{n=-\infty}^{\infty} G[n]\cdot \delta \left(f-\frac{n}{\tau}\right)
</math>
 
where variable <math>f\,</math> represents a '''continuous''' frequency domain. When variable <math>x\,</math> has units of seconds, <math>f\,</math> has units of [[hertz]]. The "teeth" of the comb are spaced at multiples (i.e. [[harmonics]]) of &nbsp;<math>1/\tau,\,</math>&nbsp; which is called the [[fundamental frequency]]. The original <math>g(x)\,</math> can be recovered from this representation by an [[Fourier transform|inverse Fourier transform]].<ref>
Formally, the inverse transform is given by''':'''
 
:<math>
\begin{align}
\mathcal{F}^{-1}\{G(f)\} &=
\mathcal{F}^{-1}\left\{ \sum_{n=-\infty}^{\infty} G[n]\cdot \delta \left(f-\frac{n}{\tau}\right)\right\}\\
&= \sum_{n=-\infty}^{\infty} G[n]\cdot \mathcal{F}^{-1}\left\{\delta\left(f-\frac{n}{\tau}\right)\right\}\\
&= \sum_{n=-\infty}^{\infty} G[n]\cdot e^{i2\pi \frac{n}{\tau} x}\cdot \mathcal{F}^{-1}\{\delta (f)\}\\
&= \sum_{n=-\infty}^{\infty} G[n]\cdot e^{i2\pi \frac{n}{\tau} x} \quad = \ \ g(x).
\end{align}
</math>
</ref> The function <math>G(f)\,</math> is therefore commonly referred to as a '''Fourier transform''', even though the Fourier integral of a periodic function is not convergent.<ref>
Since the integral defining the Fourier transform of a periodic function is not convergent, it is necessary to view the periodic function and its transform as [[Distribution_(mathematics)|distribution]]s. In this sense <math>\mathcal{F}\{e^{i2\pi \frac{n}{\tau} x}\}</math> is a [[Dirac delta function]], which is an example of a [[Distribution_(mathematics)|distribution]].</ref>
 
=== Fourier series on a square ===
We can also define the Fourier series for functions of two variables ''x'' and ''y'' in the square [-&pi;,&pi;]&times;[-&pi;,&pi;]''':'''
 
:<math>f(x,y) = \sum_{j,k \in \mathbb{Z}} c_{j,k}e^{ijx}e^{iky},</math>
:<math>c_{j,k} = {1 \over 4 \pi^2} \int_{-\pi}^{\pi} \int_{-\pi}^{\pi} f(x,y) e^{-ijx}e^{-iky}\, dx \, dy\ .</math>
 
Aside from being useful for solving partial differential equations such as the heat equation, one notable application of Fourier series on the square is in [[image compression]]. In particular, the [[jpeg]] image compression standard uses the two-dimensional [[discrete cosine transform]], which is a Fourier transform using the cosine basis functions.
 
=== Hilbert space interpretation ===
 
{{main|Hilbert space}}
 
In the language of [[Hilbert space|Hilbert spaces]], the set of functions <math>\{ e_n = e^{i n x},n\in\mathbb{Z}\}</math> is an [[orthonormal basis]] for the space <math>L^2([-\pi,\pi])</math> of square-integrable functions of <math>[-\pi,\pi]</math>. This space is actually a [[Hilbert space]] with an [[inner product]] given by''':'''
 
:<math>\langle f, g \rangle \ \stackrel{\mathrm{def}}{=} \ \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x)\overline{g(x)}\,dx.</math>
 
The basic Fourier series result for Hilbert spaces can be written as
 
:<math>f=\sum_{n=-\infty}^{\infty} \langle f,e_n \rangle e_n.</math>
 
This corresponds exactly to the complex exponential formulation given above. The version with sines and cosines is also justified with the Hilbert space interpretation. Clearly, the sines and cosines form an [[orthonormal set]]:
 
:<math>\int_{-\pi}^{\pi} \cos(mx)\, \cos(nx)\, dx = \pi \delta_{mn},</math>
:<math>\int_{-\pi}^{\pi} \sin(mx)\, \sin(nx)\, dx = \pi \delta_{mn}</math>
 
(where <math>\delta_{mn}</math> is the [[Kronecker delta]]), and
 
:<math>\int_{-\pi}^{\pi} \cos(mx)\, \sin(nx)\, dx = 0.</math>
 
The density of their span is a consequence of the [[Stone-Weierstrass theorem]].