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

Note:
any function f �x � can be separated into its odd and even parts, o � x� and e � x � :
with e � x �=e �−x � , o � x�=−o�−x � we have:
f � x� = e� x ��o �x �
f �−x� = e� x �−o �x �
The Fourier Transform and its Applications, Jürgen Stutzki, Sommersemester 2007
, and hence e �x � = 1
2
similarly: separation into real and imaginary part:
f �x� = ℜ f � x��i ℑ f � x�
f * �x� = ℜ f � x�−i ℑ f � x�
o � x � = 1
2
� f �x �� f �−x��
.
� f �x �− f �−x��
, and ℜ f � x� = 1
2 � f � x�� f * � x��
ℑ f � x� = 1
2i � f � x�− f * � x��
With this, we can separate the hermitean and anti-hermitean parts:
so that
f �x �=e � x ��o� x �=ℜe � x��i ℑ e � x ��ℜ o� x ��i ℑo � x �
=[ ℜe � x ��i ℑo � x �]
hermitean
anti−hermitean
�[ ℜo �x ��i ℑ e � x� ]
=h� x ��a � x �
f * �−x �= =[ ℜe � x �−�−�i ℑ o �x �]�[−ℜ o� x �−i ℑ e �x �]=h � x �−a �x �
h� x� = 1
2 � f � x�� f * �−x��
a� x� = 1
2 � f � x�− f * �−x��
math_ground_7.odt
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