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I have come across the function

$f(t) = \sum_{j=1}^n c_j e^{2 \pi i a_j t}$

with $c_j \in \mathbb{C}$, $c_j\neq 0$ and $a_j\in\mathbb{R}$, $a_j \neq 0$ for $j=1,...,n$, and the $a_j$ distinct. I want to show that $f(t)$ is periodic with least period equal to $1/\gcd a_j$ if the $a_j$ have a common divisor, and not periodic otherwise.

Is anyone aware of a good reference for this question?

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Not really a research question, as per the faq, is it? –  Gerry Myerson Jan 5 '12 at 22:23

1 Answer 1

up vote 2 down vote accepted

For functions of this form, define an inner product by $\langle f,g\rangle=\lim_{T\to\infty}\frac1T\int_0^T f(t)\bar g(t)\,dt$. With this inner product, the set of functions $e^{2\pi i at}$ form an uncountable orthogonal set.

If $f(t)$ is periodic with period $s$, then $f(t)=f(t+s)=\sum_{j=1}^n (e^{2\pi i a_js}c_j)e^{2\pi i a_jt}$. Since the inner product of $f$ with itself is $\sum |c_j|^2$, we deduce that $e^{2\pi i a_js}=1$ for each $j$, proving the claim.

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