Classification of Fourier summation formulas on a horizontal strip

We classify Fourier summation identities in which the measure on the Fourier side is supported in a horizontal strip of $\mathbb{C}$. Let $μ=ν+η$, where $ν$ is a strongly tempered measure on $\mathbb{R}$, $η=\sum_{m\geq1} b(γ_m)δ_{γ_m}$ is a strongly tempered pure point measure supported off the rea