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Assume that $f$ is in $L^1 (\mathbb{R})$ and $g(x)= e^{2iπx}$. Compute $f * g$

I just need a hint and not the entire answer. How can I compute the convolution when I don't know what $f$ is?

Tunococ
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user311363
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    You can't. The best you can do is write the answer in terms of the Fourier transform, however that is just a different way to write the integral (and isn't actually a real simplification). – Cameron Williams Feb 07 '16 at 02:41
  • Just write down the integral which represents the convolution of the two functions!! – Mhenni Benghorbal Feb 07 '16 at 03:22

1 Answers1

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$$(f*g)(x) = \int_{-\infty}^{\infty} f(t) e^{2\pi i(x-t)} dt $$

You can simplify the above as

$$(f*g)(x) = e^{2\pi i x} F(2\pi) $$

Where $F(w) $ is the Fourier transform of $f$

$$ F(w) = \int_{-\infty}^{\infty} f(t) e^{-iwt} dt $$

Mhenni Benghorbal
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