# Limit of $\int_{-\infty}^{\infty} f(x)\sin(tx)dx$ as $|t|\to\infty$

Let $f$, a Lebesgue integrable function in $\mathbb{R}$ ($\int_{\mathbb{R}}|f| < \infty$). Let: $$g(t) := \int_{-\infty}^{\infty} f(x)\sin(tx)dx$$ Show that $g$ is continuous (which I did), and that: $$\lim_{|t| \rightarrow \infty} g(t) = 0$$

Why is the second part correct?

Thanks!

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– David Mitra Feb 25 '12 at 17:55

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This is so called Riemann-Lebesgue Lemma! – беркай Mar 20 '12 at 20:58

## 1 Answer

The result is trivial if $f$ is the characteristic function of an interval, and therefore also if it is a step function. Can you take it the rest of the way from there?

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