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This is a revision question I've been working on.

Show that if a $2\pi$-periodic function $f$ has the complex Fourier coefficients $c_{k}$ and $g(t)=f(t+a)$, where $a$ is a constant, the the Fourier coefficients $y_{k}$ of $g$ and given by $y_{k}=e^{ika}c_{k}$.

Now suppose $f$ has Fourier coefficients $c_{k}=e^{-k^{2}}$ and $g$ has Fourier coefficients $p_{k}=(1+k^{2})^{-1}$. Define $h(t)=2f(t+1)-g(t-2)-3$. Find the complex Fourier coefficients of $h$.

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What have you tried? I mean, there is not much more to do here than plugging in the definitions. – t.b. Jul 18 '11 at 16:52
up vote 2 down vote accepted

We have, from definitions, $$ c_k = \frac{1}{2\pi}\int_{-\pi}^\pi f(t)e^{ikt}dt.$$

What happens if you substitute $t + a$ for $t$ in this formula? Recall that $e^{(a+b)} = e^ae^b$.

For the next part, you can use some facts you now know to find $2f(t+1)$ and $-g(t-2).$ It just remains to find the complex Fourier coefficients of 3 (hint: this will not be hard!)

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