Fourier series

Fourier Series allow periodic functions to be broken down into linear combinations sine and cosine waves. Consequently they are extremely important in many areas of applied mathematics, particularly in frequency analysis. In many situations they can be used to solve differential equations, such as the wave equation.

Definition

Let f(x) be a real periodic square integrable function with period 2L. The Fourier Series is defined as

            inf               inf
f(x) = a0 +  Σ ancos(nπx/L) +  Σ bnsin (nπx/L).
            n=1               n=1

where the coefficients are determined by the integrals

an = 1/L integral(f(x)cos(nπx/L)dx, x=0...2L)
bn = 1/L integral(f(x)sin(nπx/L)dx, x=0...2L).

Justification

The Fourier series works because the functions cos(nπx/L) and sin(nπx/L) form a complete orthogonal set of the space of square integrable functions on the interval from 0 to 2L. We have the orthogonality relations for n,m>0 of the form

integral(sin(nπx/L)sin(mπx/L)dx, x=0...2L) = L δmn
integral(cos(nπx/L)cos(mπx/L)dx, x=0...2L) = L δmn
integral(cos(nπx/L)sin(mπx/L)dx, x=0...2L) = 0

where δ_mn is the Kronecker Delta. Now assume we have an expression of the form as above, and we wish to determine the coefficient b_m. Multiply the relation by sin(mπx/L) to obtain

                                inf                         inf
f(x) sin(mπx/L)= a0sin(mπx/L) +  Σ ancos(nπx/L)sin(mπx/L) +  Σ bnsin (nπx/L)sin(mπx/L).
                                n=1                         n=1

We now integrate both sides from 0 to 2L. By the orthogonality relations, all the integrals vanish except for the term involving b_m. Hence

integral(f(x)sin(mπx/L)dx, x=0...2L) = L bm

which gives the required result. The a_m's can be determined in a similar way by multiplying through by cos(mπx/L) and integrating.

Example

Consider a sawtooth wave, given by f(x)=x on the open interval (-L,L), which is periodic with period 2L.

        f(x)
         |
   /     |   /         /
  /      |  /         /
 /       | /         /
/        |/         /
----+----+----+----+---- x
    -L  /|    L   /2L
       / |       /
      /  |      /
     /   |     /
         |

Then by above we have

an = 1/L integral(x cos(nπx/L)dx, x=-L...L)
   = 0,

by symmetry. Using integration by parts we obtain

bn = 1/L integral(x sin(nπx/L)dx, x=-L...L)
   = 2/L integral(x sin (nπx/L)dx, x=0...L)
   = 2/L (-L/(nπ))(Lcos(nπL/L) - Lcos(nπ0/L)) + 2/(nπ) integral(cos(nπx/L),x=0...L)
   = -2L/(nπ)cos(nπ)
   = 2L/(nπ) (-1)n+1.

From this we deduce that

      inf
f(x) = Σ 2L/(nπ) (-1)n+1 sin (nπx/L)
      n=1
     
     = 2L/π (sin(πx/L) - sin(2πx/L)/2 + sin(3πx/L)/3 - sin(4πx/L)/4 + ... )

Even for a simple example such as this we have derived a remarkable result: not the sort thing that one could guess.

Complex representation

If the input function f(x) is complex, then we have an alternative formulation

       inf
f(x) =  Σ cn exp(inπx/L)
      n=-inf

where the coefficients are determined by

cn = 1/(2L) integral(f(x)exp(-inπx/L)dx, x=-L...L).

Just a few comments about convergence. In general square integrability will not guarantee convergence of the Fourier series to the original function. For example, in the case of the sawtooth mentioned above, the Fourier series will take on the value 0 at x= L and x= -L, both of which are points of discontinuity.

Convergence of the Fourier series is rather unlike convergence of a power series. A power series converges in a disk. In contrast the convergence of a fourier series is a local phenomenon. So, whether or not the fourier series converges depends on the local properties of the function at that point. In particular a sufficient condition for the fourier series to converge to the original function is that the function satisfy a Lipshitz condition at that point. This means that abs(f(x)-f(y)) can be made smaller that const*abs(x-y) in some interval about x.(Essentially a Lipshitz condition is stronger than continuity but weaker than differentiability).

A closely related subject is that of Fourier transforms.