I'm struggling with this problem. " The function $f \in L^1(\mathbb R) \cap C_0(\mathbb R)$ holds the condition : $A > 0$ exists such that $|\hat f(\alpha)| \leq A \frac{1}{|\alpha|}$, $\alpha \in \mathbb R \setminus \{0\} $.
Then for every $x \in \mathbb R$, the equality $f(x) = lim_{R \rightarrow \infty} \frac{1}{2\pi} \int^R_{-R}\hat f (\alpha)e^{i\alpha x} d\alpha $ holds. "
I already studied that if $f \in L^1(\mathbb R)$, $\hat f \in L^1(\mathbb R) $ , then the inverse Fourier transformation is possible. But, in this case, I can't ensure that $\hat f$ holds this condition, moreover, in this problem $\hat f(\alpha)e^{i\alpha x}$ may not be $L^1$-integrable(Lebesgue). Then, how can I prove this not using "inverse Fourier"?
(c.f The consequent problem of this problem is : if $f$ is first-differentiable, and $f'\in L^1(\mathbb R) \cap C_0(\mathbb R)$, the same equality holds. (In fact, I couldn't solve this problem, too.) Could you also explain whether "differentiation" is related to the solution of the above problem?)