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I have a function, f, in $L_1(R)$. I need to establish the following fourier transforms and I don't know how to do so. Can anyone guide me through one of these that would then help me get the other ones?

  1. If $ g(x) = f(x + h)$ then $\hat g(\xi) = \hat f(\xi)e^{2\pi ih\xi}$ for any $h \in R$

  2. If $g(x) = f(\delta x)$ then $\hat g(\xi) = \delta^{−1} \hat f (\delta^{−1} \xi)$ for any $\delta > 0$.

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    $\begingroup$ Try writing down the definition of $\hat f(x)$, $\hat f(x+h)$ and $\hat f(\delta x)$ for a start? PS: the Latex for "hat" is ... \hat{.} $\endgroup$ – Alexandre Halm Mar 26 '15 at 13:39
  • $\begingroup$ @AlexHalm can you point me in a direction where I can find these definitions? Unfortunately, I do not have a fourier textbook. $\endgroup$ – samsonite Mar 26 '15 at 14:02
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    $\begingroup$ Wikipedia ? Google ? $\endgroup$ – Alexandre Halm Mar 26 '15 at 14:03
  • $\begingroup$ @AlexHalm I got both of these figured out. Now my question is (for my own personal curiosity), how would these change if the function was in $L_1(R^2)$ rather than $L_1(R)$? $\endgroup$ – samsonite Mar 26 '15 at 14:42

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