Fourier transform of an $L^1$ function is uniformly continuous
$\def\Rbb{\mathbf{R}}$$\def\Cbb{\mathbf{C}}$$\def\intw{\int_{\Rbb^n}}$If $f\in L^1(\Rbb^n)$, denote the Fourier transform of $f$ as $$ \hat{f}(x) = \int_{\Rbb^n}f(t)e^{-2\pi x\cdot t}\ dt $$
Problem. If $f\in L^1(\Rbb^n)$, show that $\hat{f}:\Rbb^n\to\Cbb$ is uniformly continuous.
Note. This is a well-known theorem in the basic $L^1$ theory of Fourier transforms. It is the very first theorem ("Theorem 1.1") in Stein-Weiss's Introduction to Fourier Analysis on Euclidean spaces. The authors give the theorem at the beginning without a proof and simply say that it is "obvious". As an exercise in real analysis, I will write my own answer below. There are different ways to approach it; the essential tool is the Dominated Convergence Theorem, which allows one to exchange the limit and integral.
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Following the definition of uniform continuity, one basically needs to estimate: $$ \begin{align} |\hat{f}(x)-\hat{f}(y)| &= |\int_{\Rbb^n}f(t)(e^{-2\pi i x\cdot t}-e^{-2\pi i y\cdot t})\ dt|\\ &\le \intw |f(t)||e^{-2\pi ih\cdot t}-1|\ dt, \end{align} $$ where $h=x-y$.
If one can show that the integral on the right converges to zero as $h\to0$, then the uniform continuity follows.
It suffices to show that one can take the limit under the integral sign: $$ \lim_{h\to 0}\intw |f(t)||e^{-2\pi ih\cdot t}-1|\ dt = \intw \lim_{h\to 0}|f(t)||e^{-2\pi ih\cdot t}-1|\ dt\tag{*} $$ But that follows from the dominated convergence theorem (DCT) since the integral is dominated by $2\|f\|_1$ (by an easy application of triangle inequality on the exponential term): $$ \intw |f(t)||e^{-2\pi ih\cdot t}-1|\ dt\le \intw 2|f(t)|\ dt $$
Note. The usual version of DCT is written in terms of sequences. One can get the version for limit in a continuous variable using Heine's sequential characterization for the limit of functions. In order to show (*), it suffices to show that for every sequence $h_n$ with $h_n\to 0$, one has
$$ \lim_{n\to \infty}\intw |f(t)||e^{-2\pi ih_n\cdot t}-1|\ dt = \intw \lim_{n\to \infty}|f(t)||e^{-2\pi ih_n\cdot t}-1|\ dt $$0 comment threads
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