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2280 Lecture Record Week 15 S2016

Last Modified: April 12, 2016, 12:30 MDT.    Today: November 24, 2017, 07:51 MST.

Week 15: Sections 9.1, 9.2, 9.3, 9.4

 Edwards-Penney, sections 9.1 to 9.4
  The textbook topics, definitions and theorems
Edwards-Penney BVP 9.1 to 9.4 (0.0 K, txt, 31 Dec 1969)

Monday and Tuesday: Fourier Series. Section 9.1, 9.2, 9.3

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:   = 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 VECTOR 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  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] = / = integral(f*f[n],a..b) / integral(f[n]*f[n],a..b)

EXAMPLE 1. 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

   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 2. 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

 DEF. Piecewise continuous. Piecewise smooth.
 DEF. Periodic function of period 2L. Half-period L.

  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.

9.3 Fourier sine and cosine series.

DEF. Even function. Odd function.

THEOREM. (odd)(odd)=(even)

THEOREM. On a symmetric interval [-L,L]:
           1. Integral (odd) = 0
           2. Integral (even) = 2 * Integral over [0,L]

DEF. The Fourier Cosine series of f(t) defined only on [0,L] is
     the full Fourier series on [-L,L] of the even exension of f(t) to 
     the interval [-L,L]. The series has only cosine terms.

DEF. The Fourier Sine series of f(t) defined only on [0,L] is
     the full Fourier series on [-L,L] of the odd exension of f(t) to 
     the interval [-L,L]. The series has only sine terms.

EXAMPLE 1. Let f(t)=t on [0,L]. Find the Fourier Sine and Cosine series
           of f(t).

EXAMPLE 2. Find a formal Fourier series solution x(t) for the periodic
           BVP x'' + 4x = 4t, x(0)=x(1)=0. Choose the Fourier series for
           the interval [0,1] so that the boundary conditions are
           automatically satisfied for every term of the Fourier series.

Answer: The exact solution is x(t) = t + c*sin(2t) 
        with c chosen to make x(1)=0: c = -1/sin(2).
        The Fourier series derived for x(t) is 
               x(t) = sum(b[n]*sin(n Pi t),n=1..infinity)

THEOREM. Term-wise integration and differentiation of Fourier series.

    1. Term-by-term integration usually succeeds, because f(t) only
       needs to be piecewise continuous.

    2. Term-by-term differentiation generally fails. It works in the 
       limited setting where f(t) is continuous and f'(t) is piecewise
       continuously differentiable. There are a few other exceptions,
       not covered in Edwrds-Penney.

   EXAMPLE. The Fourier series for f(t)=t on [-L,L] converges to f(t), 
            but its term-by-term derivative diverges. The problem is 
            caused by the discontinuities of f(t).

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