The Poisson summation formula states if $f: \mathbb{R} \to \mathbb{R}$ then $\displaystyle \sum_{n \in \mathbb{Z}} f(n) = \sum_{n \in \mathbb{Z}} \hat{f}(n) $ where $$\hat{f}(\xi) = \int_{\mathbb{R}}dx \; e^{-2\pi i x \xi}\;f(x) $$

The fine print is that $f$ needs to be Schwartz class or be a tempered distribution such as the Dirac-delta function

$$ \sum_{n \in \mathbb{Z}} \delta_n(x) = \sum_{n \in \mathbb{Z}} e^{2\pi i n x} $$

It then says the Dirac comb is its own Fourier transform.

Are there counterexamples where the left and right sides converge yet these traces do not agree?

**Edit** As discussed in the comments, it appears in Katznelson's textbook although I don't understand his example very well - convolving a function with the Fejer kernel many times at different scales.

**Edit** Can anyone fill in details of Lucia's response?

2more comments