Definitions, Theorems, Examples, Topics for Asmar 2nd edition
Chapter 2: Fourier Series, Sections 2.1 to 2.10

 2.1 Periodic Functions
====
 DEF. Periodic function, fundamental period, base interval, periodic
      extension
 DEF. Piecewise continuous, piecewise smooth
 THEOREM 1. If f(x) is continuous and T-periodic, then for all real 
            numbers a,
               int(f(x),x=0 .. T) = int(f(x), x=a .. a+T)
 EXAMPLE. Base interval [0,2] with base function f(x)=1-x. Create a
          graphic for the 2-periodic extension g(x) on the whole real 
          line and find a formula for g(x) in terms of f(x).
          ANSWER: g(x)=f(x-2floor(x/2))
   MAPLE GRAPHIC
     f:=x->1-x;g:=x->f(x-2*floor(x/2));
     plot(f(x),x=0..2);plot(g(x),x=-4..6,discont=true);

 EXAMPLE. For the f(x) defined in the example, with g(x) its 2-periodic 
          extension, compute 
            (a) int(g(x)^2,x=-1..1); 
            (b) int(g(x)^2,x=-N..N)
    ANSWER: (a) equals int(f(x)^2,x=0..2)=2/3
            (b) equals N*(2/3)

 Orthogonality of the trigonometric system
 DEF. Orthogonal functions, orthogonality relations
 Exercise 2.1-8. cos(x)+cos(Pi x) is not periodic. See the HW set.

 2.2  Fourier Series
====
 DEF. Fourier series
 DEF. Euler coefficient formulas
 DEF. Nth partial sum of a Fourier series
 EXAMPLE. Sawtooth wave. Base function  f0(x)=(Pi-x)/2 on [0,2 Pi],
          f(x)=f0(x-2*Pi*floor(x/(2*Pi))) [2Pi-periodic extension]
          Compute the Fourier series
                   f(x)=sum(sin(n*x)/n,n=1..infinity)
 DEF. Gibbs phenomenon.
 THEOREM 1. Fourier series representation
   A Fourier series for f(x) converges to (f(x+)+f(x-))/2, provided
   f(x) is a 2Pi-periodic piecewise smooth function.
 EXAMPLE. Triangular wave. It is 2Pi-periodic function g(x), built from
          the base function g0(x)=x+Pi on [-Pi,0], g0(x)=Pi-x on [0,Pi].
          g(x)=Pi/2 + sum((4/(Pi*n^2))*cos(n*x),n=1..infinity,n odd)
 EXAMPLE. A series formula for Pi. Take x=0 in the formula above.
 EXAMPLE. Linear combinations of fourier series.
          Let h0(x)=(Pi-x) pulse(x,0,Pi) and extend it to h(x) as
          a periodic function of period 2Pi. Compute h=f+g/2 from 
          preceding examples.

          GIBBS. This example has discontinuities and we see Gibb's 
          overshoot behavior at multiples of Pi.

 EXAMPLE. Changing variables. Reflections and translations. Compute
          k(x)=h(-x-Pi). The square wave formula reproduced by trickery.

 2.3 Fourier series for 2p-periodic functions
====
 DEF. Euler coefficient formulas for 2p-periodic functions
 THEOREM 1. Let f(x) be 2p-periodic piecewise smooth. The Fourier series 
            of f(x) converges pointwise to the value (f(x+)+f(x-))/2.
 EXAMPLE. Fourier series of f(x)=|x| on [-p,p] (a triangular wave).
          Because the periodic extension is continuous, the Fourier 
          series converges to f(x) on |x| <= p. Its a cosine series with 
          coefficients a[0]=p/2, a[n]=0 for n even and a[n]=-4p/(Pi*n)^2
          for n odd.

 EXAMPLE. Triangular wave of period 2p and amplitude a. Change variables 
          on the previous example, g(x)=a - (a/p)f(x), which is a cosine
          series with coefficients a[0]=a/2, a[n]=0 for n even,
          a[n]=4a/(n*Pi)^2 for n odd.
 
 EXAMPLE. Manipulations involving the period and base interval.

 DEF. Even function, odd function.
 THEOREM. int(f(x),x=-p..p) = 2*int(f(x),x=0..p) if f(x) is even
 THEOREM. int(f(x),x=-p..p) = 0 if f(x) is odd
 THEOREM. (Even)(Even)=(Odd)(Odd)=Even, (Even)(Odd)=Odd.
 THEOREM 2. 
   An even function is a cosine series, all coefficients a[n]=0.
   An odd function is a sine series, all coefficients b[n]=0.

 EXAMPLE. Fourier series of an even function f(x)=1-x^2 on |x|<1.
          Ans: a[0]=2/3, a[n]=-4(-1)^n/(Pi*n)^2

 EXAMPLE. Fourier series of an odd function f(x)=x cos(x) on |x|<Pi/2.
          ANS: b[n]=(16(-1)^(n+1)/Pi)(n/((2n+1)^2(2n-1)^2))

 2.4  Half-range expansions, sine and cosine series
====
 DEF. A HALF-RANGE EXPANSION is a Fourier series representation of a 
      non-periodic function f(x) defined on 0<x<p. 
 DEF. An EVEN PERIODIC EXTENSION is an extension F(x) of f(x) from 0<x<p
      to the whole real line as a 2p-periodic function. The formula is
            f0(x)=f(x) on 0<x<p,
            f0(x)=f(-x) on -p < x < 0,  then f0 is the base function,
            F(x)=f0(x-2p floor(x/(2p)))
 DEF. An ODD PERIODIC EXTENSION is an extension F(x) of f(x) from 0<x<p
      to the whole real line as a 2p-periodic function. The formula is
            f0(x)=f(x) on 0<x<p,
            f0(x)=-f(-x) on -p < x < 0,  then f0 is the base function,
            F(x)=f0(x-2p floor(x/(2p)))
 THEOREM 1. Let f(x) be piecewise smooth on 0<x<p. Then 
    (a) (f(x+)+f(x-))/2=a[0]+sum(a[n] cos(n Pi x/p),n=1..infinity)
    (b) (f(x+)+f(x-))/2=sum(b[n] sin(n Pi x/p),n=1..infinity)
    where a[n], b[n] are coefficients given by the Euler formulas    
                a[0]=int(f(x),x=0..p)/p,
                a[n]=int(f(x)cos(n Pi x/p),x=0..p)*(2/p),
                b[n]=int(f(x)sin(n Pi x/p),x=0..p)*(2/p).
 EXAMPLE 1. Half-range expansions for f(x)=x on 0<x<1.
   ANSWER: b[n]=2(-1)^(n+1)/(n Pi), 
           a[0]=1/2, a[n]=(-4/Pi^2)/n^2 for n odd.
 EXAMPLE 2. Half-range expansion for f(x)=sin(x) on 0<x<Pi.
   ANSWER: Because the even extension is |sin(x)| and the odd extension
           is sin(x), then only the Fourier series for |sin(x)| is to be
           calculated, giving a[0]=2/Pi, a[n]=(-4/Pi)/(n^2-1) for n even
           and n>1.