I have seen some references, which indicate that Fourier coefficients of a function $f$, denoted by $c_n(f)$, have some connections with the sequence $\{\frac{1}{n}\}$. Like here and here, it seems that $c_n(f) \le \text{const}\, \frac{1}{n}$.
Are there any known results related to this question? Fourier coefficients must have been well-studied in many cases, but it is not my expertise.
If $f$ is of bounded variation than the inequality is valid. Ref. Fourier series by Edwards.