Let $\psi(x)$ be a smooth function with compact support. Let $f$ be a real valued function defined on $\mathbb{R}^n$. Then the Poisson summation formula holds $$ \sum_{x \in \mathbb{Z}^n} \psi(x) e^{2 \pi i f(x)} = \sum_{x \in \mathbb{Z}^n} \int_{\mathbb{R}^n} \psi(y) e^{2 \pi i f(y) - 2 \pi i x.y} dy $$ for suitable choice of $f$ for example if $f$ is smooth. I was wondering, does Poisson summation formula still hold if $f$ was piecewise smooth?
Furthermore, can one do most harmonic analysis with phase function with piecewise smooth function? or does it cause a serious problem in general? Any input appreciated, thank u.
A definition of piecewise smooth function on $\mathbb{R}^n$ can be found in the link below. What I had in mind was let $h: \mathbb{R} \to \mathbb{R}$ be a piecewise smooth function on $\mathbb{R}$ and let $g$ be a smooth function on $\mathbb{R}^n$. I was mostly curious about functions of the form $f = g \circ h$.