The notion of conditional convergence can be extended to integrals.
Can it also be extended to distributions - specifically for tempered distributions?
The motivation behind this question comes from looking at the Fourier transform of Dirichlet series. For instance, the Dirichlet eta function is
$\eta(s) = \sum_{n=1}^\infty \left[ (-1)^{n+1} \frac{1}{n^s} \right]$
Letting $s = \sigma + it$, This function can be rewritten as
$\eta(s) = \sum_{n=1}^\infty \left[ (-1)^{n+1} \frac{1}{n^\sigma} e^{-it \ln n}\right]$
This sum is conditionally convergent in the critical strip $0 < \sigma < 1$. We can take its Fourier transform to obtain
$\hat\eta(s) = \sum_{n=1}^\infty \left[ (-1)^{n+1} \frac{1}{n^\sigma} \delta_0(t + \ln n)\right]$
There is a certain sense in which this distribution, itself, is conditionally convergent. For instance, one function in the Schwartz space is $\phi(x) = e^{-log(x^2)^2}$. Then $\int \hat\eta(0.5+it) \phi(t) \text{dt}$ converges, whereas $\int \mid\hat\eta(0.5+it)\mid \phi(t) \text{dt}$ does not.
Hence, there is a meaningful sense in which this distribution is, itself, conditionally convergent.