  We revisit analytical methods for constraining the nonperturbative $S$-matrix
of unitary, relativistic, gapped theories in $d \geq 3$ spacetime dimensions.
We assume extended analyticity of the two-to-two scattering amplitude and use
it together with elastic unitarity to develop two natural expansions of the
amplitude. One is the threshold (non-relativistic) expansion and the other is
the large spin expansion. The two are related by the Froissart-Gribov inversion
formula. When combined with crossing and a local bound on the discontinuity of
the amplitude, this allows us to constrain scattering at finite energy and spin
in terms of the low-energy parameters measured in the experiment. Finally, we
discuss the modern numerical approach to the $S$-matrix bootstrap and how it
can be improved based on the results of our analysis.
