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Fourier series proof

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Poirot

Banned
Feb 15, 2012
250
Let f(k) be the fourier transform of F(x). Prove that the fourier transorm of F(ax) is $\frac{1}{a}f(\frac{k}{a})$ where a>0 and the fourier transform is defined to have a factor of 1/2pi.
 

CaptainBlack

Well-known member
Jan 26, 2012
890
Let f(k) be the fourier transform of F(x). Prove that the fourier transorm of F(ax) is $\frac{1}{a}f(\frac{k}{a})$ where a>0 and the fourier transform is defined to have a factor of 1/2pi.
The particular form of the FT you are using is not that important, the basic idea is that:

\[\mathfrak{F} \big[ F(ax) \big] (k) = \int_{-\infty}^{\infty} F(ax) e^{-kx{\rm{i}}}\; dx\]

Now make the change of variable \(x^{*}=ax\) and the result drops out.

CB
 

CaptainBlack

Well-known member
Jan 26, 2012
890
Let f(k) be the fourier transform of F(x). Prove that the fourier transorm of F(ax) is $\frac{1}{a}f(\frac{k}{a})$ where a>0 and the fourier transform is defined to have a factor of 1/2pi.
There is confusion here about the thread title (Fourier series) and content (Fourier transform). I have answered for the FT, is that what you intended?

CB