Chapter 9 Edwards-Penney BVP textbook
Fourier Series Methods

9.1 Periodic Functions and Trigonometric Series

DEF. Periodic Function. f(t+p)=f(t) for all t. p=period.

Orthogonality Relations

DEF.  Two functions u, v are said to be othogonal on [a,b] provided integral(u*v,a..b)=0.
      A list of functions is said to be othogonal on [a,b] provided any two of them are orthgonal on [a,b].

THEOREM. The trigonometric list of sin(nt), cos(mt), n=1..infinity, m=0..infinity, is orthogonal on [-Pi,Pi].
         The trigonometric list is independent on [-Pi,Pi], because it is a list of Euler atoms.


DEF. A Fourier series is a formal sum of trigonometric terms from the trig list.
     A Fourier sine series is a Fourier series with no cosine terms.
     A Fourier cosine series is a Fourier series with no sine terms.

Fourier Coefficient Formulas

  Let f(x) be defined on [-Pi,Pi]. Define

   a[m] = (1/Pi)*integral(f(t)*cos(mt),-Pi..Pi), m=0..infinity

   b[n] = (1/Pi)*integral(f(t)*sin(nt),-Pi..Pi), n=1..infinity

Classical Fourier Series

   (1/2)*a[0] + SUM( a[m]*cos(m*x), m=1..infinity)  +  SUM( b[n]*sin(n*x), n=1..infinity)

THEOREM. The formulas for a[m], b[n] need not be memorized. They arise from one idea:

        (1)  Start with f(x) = trigonometric series 
        (2)  Multiply the equation in (1) by one trigonometric atom. Integrate over [-Pi,Pi].
        (3)  Orthogonality implies that the integrated series has exactly one nonzero term!
             Divide to find the corresponding coefficient a[m] or b[n].

DEF:  <u,v> = integral(u*v,a..b). It has these INNER PRODUCT properties. The vector space V
      together with these properties is called an INNER PRODUCT SPACE.

         (1) <u,v+w> = <u,w> + <v,w>  linear in the first argument
         (2) <c*u,v> = c*<u,v>
         (3) <u,v>=<v,u>  symmetry
         (4) <u,u> = 0 if and only if u=0

DEF:  On inner product space V, the NORM is defined by |u| =sqrt(<u,u>), or equivalently, |u|^2 = <u,u>.

DEF: A VECTOR is a package in a set V. Set V, called a VECTO SPACE, is equipped with addition and scalar multiplication, such that the two closure laws hold and the 8 properties are valid (group under addition, scalar disribution laws).

THEOREM. Let <f[n]> n=1..infinity be a list of orthogonal functions on [a,b]. 
         Let f = SUM(c[n]*f[n],n=1..infinity).

               Then c[n] = <f,f[n]>/<f[n],f[n]> = integral(f*f[n],a..b) / integral(f[n]*f[n],a..b)

EXAMPLE. Find a[m], b[n] for the square wave f(x) = -1 on (-Pi,0), f(x) = 1 on (0,Pi), f(x)=0 for x=-Pi,0,Pi.
         Plot the Fourier series F(x) of f(x) on -2Pi to 2Pi. 

         ANSWER. a[m]=0 for all m, because f(x) is odd.
                 b[n] = 4/(n*Pi) for n odd
                 b[n] = 0 for n even
GIBB's OVERSHOOT.

   At discontinuities of f(x), F(x) has a strange behavior, called Gibb's Overshoot. This can be seen by plotting
   a truncated Fourier series near discontinuities of f(x). 

EXAMPLE.  Find the Fourier series of f(x) on [-Pi,Pi], where f(x) = x*pulse(x,0,Pi) except that f(x)=Pi/2 at x=Pi and x=-Pi.
          ANSWER.  a[0] = Pi/2
                   a[m] = [(-1)^m-1]/(m^2*Pi^2)  for m=1..infinity
                   b[n] = (-1)^n(-1)/n  for n=1..infinity

9.2  Fourier Convergence Theorem

  THEOREM. Let f(x) be smooth on [-Pi,Pi] and F(x) its formal Fourier series, built with the Fourier coefficient formulas.
           Then f(x) = F(x) for all x in [-Pi,Pi]. 

  THEOREM. The convergence theorem above continues to hold if f(x) is only piecewise smooth, but the equation f(x) = F(x)
           only holds at points of continuity of f(x). At other points, there is the equation (f(x+)+f(x-))/2  =  F(x).

  THEOREM.  The series convergence is uniform if f(x) is smooth. It is not uniform for the Gibb's example.