  In this work we generalize the Bochner criterion addressing the
characteristic function, i.e., the Fourier transform, of the Glauber-Sudarshan
phase-space function. For this purpose we extend the Bochner theorem by
including derivatives of the characteristic function. The resulting necessary
and sufficient nonclassicality criteria unify previously known moment-based
criteria with those based on the characteristic function. For applications of
the generalized nonclassicality probes, we provide direct sampling formulas for
balanced homodyne detection. A squeezed vacuum state is experimentally realized
and characterized with our method. This complete framework -- theoretical
unification, sampling approach, and experimental implementation -- presents an
efficient toolbox to characterize quantum states of light for applications in
quantum technology.
