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Real Analysis : Fourier Transform for L1R

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Proof regarding a Fourier Transform for L1R.

Show 1/2pi ∫-1 -->1 (1 -|y|)e^(iyx) dy = 1/2pi [(sin x/2)/(x/2)]^2

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Text: An Introduction to Harmoni c Analysis (Fourier Series)
by Yitzhak Katznelson

The following problem is a portion of the proof about Fourier Transforms for .

Problem 1.8: Show

The first fact we need is that ei = cos  + i sin 

(If you aren't familiar with this, check out any complex analysis introductory text - I recommend Schaum's, but whatever you or your library may have.)

Second, dealing with absolute ...

Solution Summary

An equality is proven using Fourier transforms.