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We know that if $\hat f(\omega)=\frac{1}{2\pi}\int_{-\infty}^\infty f(t)e^{-i\omega t}dt$ is the Fourier transform of $f(t)$ then under some conditions, $f(t)=\int_{-\infty}^\infty \hat f(\omega)e^{+i\omega t}d\omega $ . Now, are there examples of functions, for which the Fourier transform converges ($\hat f $ exists for almost every $\omega \in \mathbb R$) but the integral "$\mathrm{p.v.}\int_{-\infty}^\infty \hat f(\omega)e^{+i\omega t}d\omega $" doesn't converge (in a set with a measure bigger than zero)? or does converge, but not to $f(t)$? (in points where $f$ is continuous, for example)

And question number two: we know that if $f \in L^1(\mathbb R)$ then $$\int_a^t f(\tau)\,d\tau = \mathrm{p.v.}\int_{-\infty}^\infty \hat f(\omega) \frac{e^{i\omega t} - e^{i\omega a} }{i\omega} d\omega$$ and if $f \in L^2(\mathbb R)$ then we even don't need the principal value (by Plancherel's Theorem). The question is, does the integral on the right converge uniformly (with respect to $t \in \mathbb R$)? If not, can you give counter examples?

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