 Finite-time spectral boundary transform

A boundary trace transformed over a finite time interval using the dispersion relation of a linear partial differential equation. For integrable trace data and a polynomial dispersion function, it is an entire function of the complex spectral parameter. Its growth depends on $\mathrm{Re}\omega (k)$; multiplying by $e^{-\omega (k)t}$ changes the integrand to the causal factor $e^{-\omega (k)(t-s)}$.