The Fourier Transform and its applications goals: present the Fourier ...

● only the behaviour of f �x� for x�0 enters into to Cosine- and Sine-Transform
● the symmetries of F C �s� and F S �s� imply that they are fully determined by their s�0
behavior
Thus, only the x�0 behavior of an arbitrary function f �x � determines its F C �s� and F S �s�
Consider a new function f H � x � :
f � x � , x �0
f H � x �={
= f � x� H �x � ,
0, x �0
obtained by "cutting-off" the negative
part of f �x � , i.e. by multiplying f �x � with the Heavyside step-function defined above.
Obviously, it has the same Cosine- and Sine transform as f �x � itself:
F S , H �s�=ST � f H � x ��=2∫ 0
∞
=2∫ 0
and similarly F C , H �s�=F C �s� .
The Fourier Transform and its Applications, Jürgen Stutzki, Sommersemester 2007
∞
f H �x � sin �2� xs� dx
f � x � sin �2� xs� dx=ST � f � x ��=F S �s�
Now, e H � x�= 1
2
2 [ f H �x�� f H �−x�]={1
and oH �x�= 1
2
2 [ f H � x�− f H �−x�]={1
1
2
− 1
2
f � x� , x�0
f �−x� , x�0
f � x� , x�0
f �−x� , x�0
math_ground_12.odt
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