# Proove the Riemann-Lebesgue lemma for the Fourier Transform on $\mathbb R$ using the Riemann-Lebesgue lemma for the Fourier coefficient on the circle

Proove the Riemann-Lebesgue lemma for the Fourier Transform on $\mathbb R$ using the Riemann-Lebesgue lemma for the Fourier coefficient on the circle.

Periodize $f \in L^1(\mathbb R)$ and $f_\epsilon(x)=\epsilon f(\epsilon x)$ with \begin{align} g(x)&=\sum_{k=-\infty}^\infty f(x+2k\pi) ,\\ g_\epsilon (x)&=\sum_{k=-\infty}^\infty f_\epsilon(x+2k\pi). \end{align} Then \begin{align} \hat g(n) &= \hat f(n) \tag{1} ,\\ \hat g_\epsilon(n) &= \hat f \left(\frac{n}{\epsilon}\right). \tag{2} \end{align}

Writing $$\hat f(x) = \hat f(x) - \hat f \left( x+\frac{n}{\epsilon} \right) + \hat f \left( x+\frac{n}{\epsilon} \right)$$

I'm not sure on how to take limits to get $\lim_{x \to \infty} \hat f(x)=0$.

I know $\hat f$ is uniformly continuous.
I'm not sure if the the Riemann-Lebesgue lemma for Fourier coefficient apply when $\epsilon \to 0$ as $x+\frac{n}{\epsilon}$ may not be an integer.

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