Chapter 8
              Fourier Transform
              We introduce the Fourier transform, a special linear integral transformation for
              differential equations which are defined on unbounded domains. The method is
              associated with the French physicist and mathematician Joseph Fourier whose
              work opened up a new way to solve linear PDEs. As we will see later, the Fourier
              transform reduces a partial differential equation in (t; x)-domain to an ordinary
              differential equation in (t;!)-domain.
              8.1 Definition
              The Fourier transform is a linear integral transformation of a function f(x) in
                                        ^
              x-domain to another function f(!) in !-domain. As we will see in sequel, ! can
                                                         ^
              be considered as the frequency variable and thus f(!) represents the frequency
              distribution embedded in function f(x).
                 1. (Admissible functions) We first introduce the admissible space of func-
                    tions.
                    Definition 8.1. A function f(x); ¡1 < x < 1 is called integrable if the
                    following relation holds Z
                                          1jf(x)jdx<1:                       (8.1)
                                         ¡1
                    Afunction f(x);¡1
                                              >p
                                         f(x)=< x 1    x>0;
                                              >
                                              > p
                                              :¡ ¡x x<0
                    is not admissible even it satisfies the following property
                                          lim Z Rf(x)dx=0;:
                                         R!1 ¡R
                                               1
               2                                                      Fourier Transform
                     Note that
                                  Z R                Z R 1           p
                               lim    jf(x)jdx=2 lim    p dx= lim 4 R=1:
                              R!1 ¡R             R!1 0    x     R!1
                     TheFouriertransformisusuallydefinedforadmissiblefunctions. However,
                     as we will see below, the definition can be extended for a wider class of
                     functions than the admissible ones.
                  2. (Definition of Fourier transform) Suppose f is a function (not neces-
                     sarily admissible) defined on R:(¡1;1). Its Fourier transform Fffg:=
                     ^
                     f(!) is defined by the following integral
                                            ^     Z 1     ¡i!x
                                   Fffg:=f(!)= ¡1f(x)e        dx;                 (8.2)
                     as long as the integral exists. The improper integral is understood in the
                     following sense
                                Z 1f(x)e¡i!xdx= lim Z Rf(x)e¡i!xdx:               (8.3)
                                 ¡1              R!1 ¡R
                     Example 8.1. Consider the following function
                                                81 00;
                                                                    ¡1 x<0
                           is not admissible, and the definition of the Fourier transform fails as
                                      ^      Z 0      ¡i!x      Z 1 ¡i!x             Z 1
                                     f(!)=        ¡e      dx+        e     dx=¡2i        sin(!x)dx:
                                               ¡1                 0                   0
                4                                                             Fourier Transform
                       Now, let us consider the following function
                                                 f (x)= e¡"x x>0 :
                                                  "          "x
                                                           ¡e    x<0
                       The function is admissible and its transform is
                             ^      Z 1        ¡i!x        Z 1 ¡"x                    !
                             f (!)=     f (x)e     dx=¡2i      e   sin(!x)dx=¡2i           :
                              "          "                                          "2+!2
                                     ¡1                      0
                                                              ^
                       Since f (x)!f(x) for "!0, we define f(!) through the following limit
                              "
                                                 ^         ^       ¡2i
                                                 f(!)=limf (!)=       :
                                                            "       !
                                                        "!0
                                                       ^
                    4. (Inverse transform) Suppose f(!) is a function defined on !2(¡1;1).
                                                          ^
                       The inverse Fourier transform of f is defined by the following integral
                                            ¡1 ^    1 Z 1 ^     i!x
                                          F (f)=2 ¡1f(!)e         d!;                      (8.4)
                       as long as the integral exists. The integral is understood in the following sense
                                     Z 1^      i!x          Z R ^     i!x
                                         f(!)e    d!= lim       f(!)e   d!:                 (8.5)
                                      ¡1               R!1 ¡R
                                                                                      ¡1
                       The following theorem establishes the relationship between F;F   . See the
                       appendix of this chapter for a proof.
                       Theorem 8.1. Assume that f is an admissible function, then
                                                           +       ¡
                                            ¡1 ^       f(x )+f(x )
                                          F (f)(x)=           2      ;                      (8.6)
                                 +          ¡
                       where f(x ) and f(x ) are the right and left limits of f at x respectively.
                                                                           ^   2(1¡cosw)
                       Example8.3. Letusverifythetheoremforfunctions f=           i!   . Wehave
                          ¡1 ^       1      Z R1¡cos! i!x         2     Z R1¡cos!
                        F (f)(x)=i lim            !     e   d!= lim          !     sin(!x)d!;
                                       R!1 ¡R                       R!1 0
                       where we used the symmetry property for the integration. The following
                       figures shows the value of the integral for R = 20; 100 respectively. It is
                       observed that for R!1 the integral approaches f(x).