Arbitrarily slow Fourier coefficient decay (${\mathrm{limsup}}_{n\to \infty }k(n)|\hat{g}(n)|=\infty$)

Given any positive $k(n)\to \infty$, choose increasing integers $n_j$ with $k(n_j)\ge j{2}^j$. The absolutely uniformly convergent series $g(t)={\sum }_j{2}^{-j}{e}^{i{n}_{j}t}$ 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.