Fourier Series

Fourier series are expansions of Periodic Functions $f(x)$ in terms of an infinite sum of Sines and Cosines

$$f(x) = \sum_{n=0}^\infty a_n'\cos(nx) + \sum_{n=0}^\infty b_n'\sin(nx).$$ (1)

Fourier series make use of the Orthogonality relationships of the Sine and Cosine functions, which can be used to calculate the coefficients $a_n$ and $b_n$ in the sum. The computation and study of Fourier series is known as Harmonic Analysis.

To compute a Fourier series, use the integral identities

$$\int_{-\pi}^\pi \sin(mx)\sin(nx)\,dx = \pi\delta_{mn} \qquad \hbox {for } n, m \not = 0$$ (2)

$$\int_{-\pi}^\pi \cos(mx)\cos(nx)\,dx = \pi\delta_{mn} \qquad \hbox {for } n, m \not = 0$$ (3)

$$\int_{-\pi}^\pi \sin(mx)\cos(nx)\,dx = 0$$ (4)

$$\int_{-\pi}^\pi \sin(mx)\,dx = 0$$ (5)

$$\int_{-\pi}^\pi \cos(mx)\,dx = 0,$$ (6)

where $\delta_{mn}$ is the Kronecker Delta. Now, expand your function $f(x)$ as an infinite series of the form

$$f(x) = \sum_{n=0}^\infty a_n'\cos(nx) + \sum_{n=0}^\infty b_n'\sin(nx)$$

$$= {\textstyle{1\over 2}}a_0 +\sum_{n=1}^\infty a_n\cos(nx) + \sum_{n=1}^\infty b_n\sin(nx),$$ (7)

where we have relabeled the $a_0=2a_0'$ term for future convenience but set $b_n=b_n'$ and left $a_n=a_n'$ for $n\geq 1$. Assume the function is periodic in the interval $[-\pi,\pi]$. Now use the orthogonality conditions to obtain

$\int_{-\pi}^\pi f(x)\,dx$
$\quad = \int_{-\pi}^\pi \left[{\sum_{n=1}^\infty a_n\cos(nx) + \sum_{n=1}^\infty b_n\sin(nx) + {\textstyle{1\over 2}}a_0}\right]\,dx$
$\quad = \sum_{n=1}^\infty \int_{-\pi}^\pi [a_n\cos(nx)+b_n\sin(nx)]\,dx + {\textstyle{1\over 2}}a_0 \int_{-\pi}^\pi dx$
$\quad = \sum_{n=1}^\infty (0+0)+\pi a_0 = \pi a_0$	(8)

and

$\int_{-\pi}^\pi f(x)\sin(mx)\,dx$
$= \int_{-\pi}^\pi \left[{\sum_{n=1}^\infty a_n\cos(nx)+\sum_{n=1}^\infty b_n\sin(nx)+{\textstyle{1\over 2}}a_0}\right]\sin(mx)\,dx$
$= \sum_{n=1}^\infty \int_{-\pi}^\pi [a_n\cos(nx)\sin(mx)+b_n\sin(nx)\sin(mx)]\,dx+{\textstyle{1\over 2}}a_0 \int_{-\pi}^\pi \sin(mx)\,dx$
$= \sum_{n=1}^\infty (0+b_n\pi \delta_{mn})+0 = \pi b_n,$	(9)

so

$$\int_{-\pi}^\pi f(x)\cos(mx)\,dx = \int_{-\pi}^\pi\left[{\sum_{n=1}^\infty a_n\cos(nx) + \sum_{n=1}^\infty b_n\sin(nx)+ {\textstyle{1\over 2}}a_0}\right]\cos(mx)\,dx$$

$$= \sum_{n=1}^\infty \int_{-\pi}^\pi [a_n\cos(nx)\cos(mx)+b_n\sin(nx)\cos(mx)]\,dx+{\textstyle{1\over 2}}a_0 \int_{-\pi}^\pi \cos(mx)\,dx$$

$$= \sum_{n=1}^\infty (a_n\pi \delta_{mn}+0)+0 = \pi a_n.$$ (10)

Plugging back into the original series then gives

$$a_0 = {1\over\pi} \int_{-\pi}^\pi f(x)\,dx$$ (11)

$$a_n = {1\over\pi} \int_{-\pi}^\pi f(x)\cos(nx)\,dx$$ (12)

$$b_n = {1\over\pi} \int_{-\pi}^\pi f(x)\sin(nx)\,dx$$ (13)

for $n=1$, 2, 3, .... The series expansion converges to the function $\bar f$ (equal to the original function at points of continuity or to the average of the two limits at points of discontinuity)

$$\bar f \equiv \cases{ {\textstyle{1\over 2}}\left[{\lim_{x\... ...to \pi_-} f(x)}\right]\cr \quad{\rm for\ } x_0 = -\pi, \pi\cr}$$ (14)

if the function satisfies the Dirichlet Conditions.

Near points of discontinuity, a ``ringing'' known as the Gibbs Phenomenon, illustrated above, occurs. For a function $f(x)$ periodic on an interval $[-L,L]$, use a change of variables to transform the interval of integration to $[-1,1]$. Let

$$x \equiv {\pi x'\over L}$$ (15)

$$dx = {\pi dx'\over L}.$$ (16)

Solving for $x'$, $x' = {Lx/\pi}$. Plugging this in gives

$$f(x') = {\textstyle{1\over 2}}a_0 + \sum_{n=1}^\infty a_n\co... ...right)+ \sum_{n=1}^\infty b_n\sin\left({n\pi x'\over L}\right)$$ (17)

$$\cases{ a_0 = {1\over L} \int^L_{-L} f(x')\,dx'\cr a_n = {... ...} \int^L_{-L} f(x')\sin\left({n\pi x'\over L}\right)\,dx'\cr}.$$ (18)

If a function is Even so that $f(x) = f(-x)$, then $f(x)\sin(nx)$ is Odd. (This follows since $\sin(nx)$ is Odd and an Even Function times an Odd Function is an Odd Function.) Therefore, $b_n = 0$ for all $n$. Similarly, if a function is Odd so that $f(x) = f(-x)$, then $f(x)\cos(nx)$ is Odd. (This follows since $\cos(nx)$ is Even and an Even Function times an Odd Function is an Odd Function.) Therefore, $a_n = 0$ for all $n$.

Because the Sines and Cosines form a Complete Orthogonal Basis, the Superposition Principle holds, and the Fourier series of a linear combination of two functions is the same as the linear combination of the corresponding two series. The Coefficients for Fourier series expansions for a few common functions are given in Beyer (1987, pp. 411-412) and Byerly (1959, p. 51).

The notion of a Fourier series can also be extended to Complex Coefficients. Consider a real-valued function $f(x)$. Write

$$f(x) = \sum_{n=-\infty}^\infty A_ne^{inx}.$$ (19)

Now examine

$\int_{-\pi}^\pi f(x)e^{-imx}\,dx = \int_{-\pi}^\pi \left({\sum_{n=-\infty}^\infty A_n e^{inx}}\right)e^{-imx}\,dx$
$= \sum_{n=-\infty}^\infty A_n \int_{-\pi}^\pi e^{i(n-m)x}\,dx$
$= \sum_{n=-\infty}^\infty A_n \int_{-\pi}^\pi \left\{{\cos [(n-m)x]+i\sin [(n-m)x]}\right\}\,dx$
$= \sum_{m=-\infty}^\infty A_n 2\pi\delta_{mn} = 2\pi A_m,$	(20)

so

$$A_n = {1\over 2\pi} \int_{-\pi}^\pi f(x)e^{-inx}\,dx.$$ (21)

The Coefficients can be expressed in terms of those in the Fourier Series

$$A_n = {1\over 2\pi} \int_{-\pi}^\pi f(x)[\cos (nx)-i\sin (nx)]\,dx$$

$$= \left\{\begin{array}{ll} {1\over 2\pi}\int_{-\pi}^\pi f(x)[\cos(n... ...i}\int_{-\pi}^\pi f(x)[\cos(nx)-i\sin(nx)]\,dx & \mbox{$n>0$}\end{array}\right.$$

$$= \left\{\begin{array}{ll} {1\over 2}(a_n+ib_n) & \mbox{$n<0$}\\ {... ... 2}a_0 & \mbox{$n=0$}\\ {1\over 2}(a_n-ib_n) & \mbox{$n>0$.}\end{array}\right.$$ (22)

For a function periodic in $[-L,L]$, these become

$$f(x) = \sum_{n=-\infty}^\infty A_ne^{i(2\pi nx/L)}$$ (23)

$$A_n = {1\over L} \int^{L/2}_{-L/2} f(x)e^{-i(2\pi nx/L)}\,dx.$$ (24)

These equations are the basis for the extremely important Fourier Transform, which is obtained by transforming $A_n$ from a discrete variable to a continuous one as the length $L\to\infty$.

See also Dirichlet Fourier Series Conditions, Fourier Cosine Series, Fourier Sine Series, Fourier Transform, Gibbs Phenomenon, Lebesgue Constants (Fourier Series), Legendre Series, Riesz-Fischer Theorem

References

Fourier Transforms

Arfken, G. ``Fourier Series.'' Ch. 14 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 760-793, 1985.

Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, 1987.

Brown, J. W. and Churchill, R. V. Fourier Series and Boundary Value Problems, 5th ed. New York: McGraw-Hill, 1993.

Byerly, W. E. An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, 1959.

Carslaw, H. S. Introduction to the Theory of Fourier's Series and Integrals, 3rd ed., rev. and enl. New York: Dover, 1950.

Davis, H. F. Fourier Series and Orthogonal Functions. New York: Dover, 1963.

Dym, H. and McKean, H. P. Fourier Series and Integrals. New York: Academic Press, 1972.

Folland, G. B. Fourier Analysis and Its Applications. Pacific Grove, CA: Brooks/Cole, 1992.

Groemer, H. Geometric Applications of Fourier Series and Spherical Harmonics. New York: Cambridge University Press, 1996.

Körner, T. W. Fourier Analysis. Cambridge, England: Cambridge University Press, 1988.

Körner, T. W. Exercises for Fourier Analysis. New York: Cambridge University Press, 1993.

Lighthill, M. J. Introduction to Fourier Analysis and Generalised Functions. Cambridge, England: Cambridge University Press, 1958.

Morrison, N. Introduction to Fourier Analysis. New York: Wiley, 1994.

Sansone, G. ``Expansions in Fourier Series.'' Ch. 2 in Orthogonal Functions, rev. English ed. New York: Dover, pp. 39-168, 1991.