An example of a trigonometric series that is not a Fourier series

Question

I am looking forward to a positive sequence of $\{a_n\}_{n\geq1}$ such that all the following points hold:

1) $\sum \frac{a_n}{n}$ is convergent.

2) There is no $f\in L^1[-\pi,\pi]$ whose Fourier series is $\sum a_n\sin nx$.

Answer 1

Let $a_n = 1$ if $\sqrt n\in\mathbb N$ and $a_n = \frac 1n$ otherwise. Then your series cannot be generated by an $L^1$-function due to the Riemann-Lebesgue-lemma.