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

Proof:
1) for the Cosine back-trafo:
define an even function for all x : f 1� x �= f �∣x∣�={
Its FT is
FT � f 1 � x ��=F 1 �s�=CT �e 1 � x��
� CT� f 1� x��
−i ST � �o1 �x ��=CT
� f � x��=F C �s�
,
=0
which is also even, being a pure cosine transform:
F 1 �−s�=F 1 �s� , resp. E 1 �s�=F 1 �s � , O 1 �s�=0
Its back-transform reproduces f 1 � x � :
f 1� x �=FT �i � F 1 �s��=CT � E 1�s��
� CT� F 1 �s��
�i ST �O 1 �s��
� =0
� �
=0
ST�F 2 �s��
The Fourier Transform and its Applications, Jürgen Stutzki, Sommersemester 2007
f � x � , x�0
f �−x � , x�0 , i.e. e 1 � x �= f 1 � x� , o 1 �x �=0 .
=CT �CT � f �∣x∣���
and for
x�0 , where f �x � is defined: f � x �= f 1 � x �=CT �CT � f � x �� �=CT � F C � s� � q.e.d.
2) similarly for the Sine backtrafo, but with the odd function f 2 �x �={ f � x � , x�0
− f �−x � , x�0 :
FT � f 2� x��=F 2�s�=CT � �e 2� x��
=0
−i ST ��o 2� x ��
ST � f 2�x�� =−i ST � f �x ��=−i F S�s� , and
f 2�x�=FT �i � F 2�s��=CT � E2�s�� �i ST �O 2�s�� =i �−i�ST � F s�s��=ST �ST � f �∣x∣�� �
q.e.d.
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