Recovering a function using the inverse fourier transform

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  • #1
hahashahid
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Homework Statement


The argument of the kernel of the Fourier transform has a different sign for the forward and inverse transform. For a general function, show how the original function isn’t recovered upon inverse transformation if the sign of the argument is the same for both the forward and inverse transform.


Homework Equations


The Fourier and the Inverse Fourier transform integrals


The Attempt at a Solution


Do I need to prove the inverse fourier theorem or is there a simpler solution?
One attempt that I have seen at recovering the original function using the inverse fourier transform formula integrates the exponent part to a sin(Mx)/x type function, which is then shown to approach the delta function as M approaches ∞.
 

Answers and Replies

  • #2
vela
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Let
$$F(\omega) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty f(t)e^{-i\omega t}\,dt.$$ The problem is simply asking you to show that if you calculate
$$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty F(\omega)e^{-i\omega t}\,d\omega,$$ you do not recover f(t). Just plug the expression for F(ω) into the second integral and calculate away.
 

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