# The decay of Fourier coefficients and the continuity of functions

Let $$f$$ be a function on $$\mathbb{T}=[0,1]$$ ($$1$$-periodic) with bounded variation. Prove that if $$\widehat{f}(k)=\int_0^1f(x)e^{-2\pi ikx}dx=o(1/|k|)$$, then $$f\in C(\mathbb{T})$$. I do not know how to use the assumption that $$f$$ is of bounded variation. Can you give me some hints?

• Can you use integration by parts for Riemann-Stieltjes integral? Sep 18, 2022 at 15:26

The (distributional) derivative $$\mu=f'$$ is a measure, and by assumption $$\widehat{\mu_n}=o(1)$$. By Wiener's theorem, this implies that $$\mu$$ does not have a point part, so $$\mu$$ is a continuous measure and thus $$f$$ is a continuous function. Compare Corollary 13.11 of my lecture notes here.