On the uniqueness of trigonometric series

It is proved that
\begin{equation} \frac {a_0}2+\sum_{n=1}^\infty(a_n\cos nx+b_n\sin nx)=\sum_{n=0}^\infty A_n(x) \end{equation}
is the Fourier series of an integrable function $f(x)$ if and only if

• 1) $\lim\limits_{h\to0}S(x,h)=f(x)$ almost everywhere, and
• 2) $\lim\limits_{\lambda\to\infty}\inf\lambda\mu\{x\in[0,2\pi]\colon S^\ast(x)>\lambda\}=0$,

where $S(x,h)=\sum\limits_{n=0}^\infty A_n(x)\biggl(\dfrac{\sin nh}{nh}\biggr)^2$ and $S^\ast(x)=\sup\limits_{h>0}|S(x,h)|$.
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