= Arbitrarily slow Fourier coefficient decay
{title2=$\limsup_{n\to\infty}k(n)|\widehat g(n)|=\infty$}
Given any positive $k(n)\to\infty$, choose increasing integers $n_j$ with $k(n_j)\geq j2^j$. The absolutely uniformly convergent series $g(t)=\sum_j2^{-j}e^{in_jt}$ is continuous, and its Fourier coefficient at $n_j$ is exactly $2^{-j}$. The displayed unbounded limsup follows, although every individual continuous function still has coefficients tending to zero by the <Riemann-Lebesgue lemma>.
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