Processing math: 61%
Google
 
Web unafbapune.blogspot.com

Thursday, January 09, 2014

 

^f(n)=(ik)nˆf

The Fourier Transform of a function f(x) can be defined as:
  ^f(x)=12πeikxf(x)dx
So,
  ^f(x)=12πeikxf(x)dxsubstitue f(x) by f(x)
Using integration by parts, let
  dv=f(x)dxv=f(x)u=eikxdu=ikeikxdx
Now,
  12πeikxf(x)dx=12πudv=12π(uv|vdu)=12π(eikxf(x)|+ikeikxf(x)dx)

Suppose f(x)0 as x±,
  12πeikxf(x)dx=ik12πeikxf(x)dx
Telescoping,
  \begin{aligned} \widehat{f^{(n)}} &= (ik)^{n} \cdot \widehat{f} \\ \end{aligned}
\Box

Source: Computational Methods for Data Analysis.


Comments: Post a Comment

<< Home

This page is powered by Blogger. Isn't yours?