If a $2\pi$-periodic function $f: \mathbb{R} \rightarrow \mathbb{R}$ is Lebesgue integrable in $[-\pi,\pi]$, and the series $\frac{a_0}{2}+\sum_{n=1}^\infty [a_n \cos{nx}+b_n \sin{nx}] $, where $(a_n), (b_n)$ are some real sequences, is convergent to $f$ uniformly or in $L_p$ norm or pointwise, then it is known that $a_n$, $b_n$ are Fourier coefficients of $f$. (The last part of this statement it is du Bois-Reymond theorem).
My question concerns analogue of such type theorems for Fourier integrals. Namely:
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be Lebesgue integrable and let
$f(x)=\lim_{T\rightarrow \infty} \int_0^T [a(\omega) \cos (\omega x)+b(\omega )\sin(\omega x)]d\omega$ for $x \in \mathbb{R}$.
Under what conditions:
$a(\omega)=\frac{1}{\pi} \int_{-\infty}^\infty f(x) \cos(\omega x)dx$,
$b(\omega)=\frac{1}{\pi} \int_{-\infty}^\infty f(x) \sin(\omega x)dx$ ?
Thanks.