Fourier series
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In mathematics, a Fourier series, named in honor of Joseph Fourier (1768-1830), is a representation of a periodic function with period 2π as a sum of periodic functions of the form
- <math>x\mapsto e^{inx},<math>
which are the harmonics of ei x. By Euler's formula, the series can be expressed equivalently in terms of sine and cosine functions. This can be generalized to periodic functions of any positive period.
Fourier was the first to study systematically such infinite series, after preliminary investigations by Euler, d'Alembert, and Daniel Bernoulli. He applied these series to the solution of the heat equation, publishing his initial results in 1807 and 1811, and publishing his Théorie analytique de la chaleur in 1822. From a modern point of view, Fourier's results are somewhat informal, due in no small part to the lack of a precise notion of function and integral in the early nineteenth century. Later, Dirichlet and Riemann expressed Fourier's results with greater precision and formality.
Many other Fourier-related transforms have since been defined, extending to other applications the initial idea of representing any periodic function as a superposition of harmonics. This general area of inquiry is now sometimes called harmonic analysis.
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Definition of Fourier series
Suppose that f(x), a complex-valued function of a real variable, is periodic with period 2π, and is square-integrable over the interval from 0 to 2π. Let
- <math>F_n =\frac{1}{2\pi}\int_{-\pi}^\pi f(x)\,e^{-inx}\,dx.<math>
Each Fn is called a Fourier coefficient. Then, the Fourier series representation of f(x) is given by
- <math>f(x) = \sum_{n=-\infty}^{\infty} F_n \,e^{inx}.<math>
Each term in this sum is called a Fourier mode or a harmonic. In the important special case of a real-valued function f(x), one often uses the equality
- <math>e^{inx}=\cos(nx)+i\sin(nx) \,\!<math>
(derived from Euler's formula) to equivalently represent f(x) as an infinite linear combination of functions of the form <math>\cos(nx) \,\!<math> and <math>\sin(nx) \,\!<math>, that is
- <math>f(x) = \frac{1}{2}a_0 + \sum_{n=1}^\infty\left[a_n\cos(nx)+b_n\sin(nx)\right]<math>, where
- <math>a_n = \frac{1}{\pi}\int_{-\pi}^\pi f(x)\cos(nx)\,dx<math> and <math>b_n = \frac{1}{\pi}\int_{-\pi}^\pi f(x)\sin(nx)\,dx<math>
which corresponds to <math>F_n = (a_n - i b_n) / 2 \,\!<math> and <math>F_n = F_{-n}^*.<math>
Example
Let f(x) = x be the identity function for x from −π to π. Outside this domain, the Fourier series implicitly requires that we define the function periodically.
We will compute the Fourier coefficients for this function. Notice that cos(nx) is an even function, while f and sin(nx) are odd functions.
- <math>a_0 = \frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)dx=\frac{1}{2\pi}\int_{-\pi}^{\pi} x dx= 0<math>
- <math>a_n=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos(nx)dx= \frac{1}{\pi}\int_{-\pi}^{\pi}x \cos(nx)dx = 0<math>
- <math> b_n= \frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\sin(nx)dx=\frac{1}{\pi}\int_{-\pi}^{\pi} x \sin(nx)dx =<math>
- <math>=\frac{2}{\pi}\int_{0}^{\pi} x\sin(nx) dx= \frac{2}{\pi}\left(
\left[-\frac{x\cos(nx)}{n}\right]_0^{\pi}+\left[\frac{\sin(nx)}{n^2}\right]_0^{\pi} \right)=(-1)^{n+1}\frac{2}{n}<math>
Notice that a0 and an are 0 because x and x cos(nx) are odd functions. Hence the Fourier series for f(x) = x is:
- <math>f(x)=x=a_0 + \sum_{n=1}^{\infty}(a_n\cos(nx)+b_n\sin(nx)) =<math>
- <math>=\sum_{n=1}^{\infty}(-1)^{n+1}\frac{2}{n} \sin(nx), \quad \forall x\in (-\pi,\pi)<math>
For an application of this Fourier series, see the value of the Riemann zeta function at s=2.
Convergence of Fourier series
While the Fourier coefficients an and bn can be formally defined for any function for which the integrals make sense, whether the series so defined actually converges to f(x) depends on the properties of f.
The simplest answer is that if f is square-integrable then
- <math>\lim_{N\rightarrow\infty}\int_{-\pi}^\pi\left|f(x)-\sum_{n=-N}^{N}
F_n\,e^{inx}\right|^2\,dx=0<math>
(this is convergence in the norm of the space L2).
There are also many known tests that ensure that the series converges at a given point x. For example, if the function is differentiable at x. Even a jump discontinuity does not pose a problem: if the function has left and right derivatives at x, then the Fourier series will converge to the average of the left and right limits (but see Gibbs phenomenon).
However, a fact that many find surprising, is that the Fourier series of a continuous function need not converge pointwise. A discussion of the counterexample, along with other positive and negative results in the general spirit of "for functions of type X, the Fourier series converges in sense Y" may be found in Convergence of Fourier series.
Orthogonality
The Fourier basis functions are orthogonal in the discrete space
- <math>\sum_{n=-\infty}^\infty e^{inx}e^{-iny}=2\pi\sum_{n=-\infty}^\infty
\delta(x-y+2\pi n)=2\pi\,\delta_{2\pi}(x-y)<math>
where δ(x) is the Dirac delta function and δT(x) is the Dirac comb function. The Fourier basis functions are orthogonal in the continuous space as well:
- <math>\frac{1}{2\pi}\int_{-\pi}^\pi e^{inx}e^{-imx}\,dx = \delta_{nm}<math>
where δnm is the Kronecker delta function.
Some positive consequences of the homomorphism properties of exp
Because "basis functions" eikx are homomorphisms of the real line (more precisely, of the "circle group") we have some useful identities:
Shifting property
If
- <math>g(x)=f(x-y) \,\!<math>
then (if G is the transform of g)
- <math>G_k = e^{-iky}F_k \,\!<math>.
Convolution theorems
- Main article: Convolution
If h(t) is the cyclic convolution of f(t) and g(t):
- <math>h(t)=\int_{-\pi}^\pi f(t')g(t-t')\,dt'<math>
where g(t)=g(t+2nπ), then the Fourier series transforms are related by:
- <math>H_n=2\pi\,F_nG_n.\,<math>
Conversely, if Hn=2πFnGn, then h(t) will be the cyclic convolution of f(t) and g(t).
In the discrete space, if Hn is the discrete convolution of Fn and Gn:
- <math>H_k=\sum_{n=-\infty}^\infty F_n G_{k-n}<math>
then the inverse transforms are related by:
- <math>h(t)=f(t)g(t)\,<math>
and conversely, if h(t)=f(t)g(t), then Hn will be the discrete convolute of Fn and Gn.
These theorems may be proven using the orthogonality relationships.
Plancherel's and Parseval's theorem
Another important property of the Fourier series is the Plancherel theorem
- <math>\sum_{n=-\infty}^\infty F_nG^*_n = \frac{1}{2\pi} \int_{-\pi}^\pi f(x)g^*(x)\,dx <math>
Parseval's theorem, a special case of the Plancherel theorem, states that
- <math>\sum_{n=-\infty}^\infty |F_n|^2 = \frac{1}{2\pi} \int_{-\pi}^\pi |f(x)|^2 dx \,<math>
which can be restated for the real-valued f(x) case above,
- <math>\frac{a_0^2}{4} + \frac{1}{2} \sum_{n=1}^\infty \left( a_n^2 + b_n^2 \right) = \frac{1}{2\pi} \int_{-\pi}^\pi f(x)^2 dx<math>.
These theorems may be proven using the orthogonality relationships.
General formulation
The useful properties of Fourier series are largely derived from the orthogonality and homomorphism property of the functions <math>e^{inx} \,\!<math>. Other sequences of orthogonal functions have similar properties, although some useful identities concerning e.g. convolutions are no longer true once we lose the homomorphism property. Examples include sequences of Bessel functions and orthogonal polynomials. Such sequences are commonly the solutions of a differential equation; a large class of useful sequences are solutions of the so-called Sturm-Liouville problems.
See also
References
- Yitzhak Katznelson, An introduction to harmonic analysis, Second corrected edition. Dover Publications, Inc., New York, 1976. ISBN 0486633314
External links
- Fourier series example problems (http://www.exampleproblems.com/wiki/index.php?title=Fourier_Series) at exampleproblems.com
- Java applet (http://www.falstad.com/fourier/) shows Fourier series expansion of an arbitrary function
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