= Finite-time spectral boundary transform
{title2=$G_j(k,t)=\int_0^t e^{\omega(k)s}g_j(s)ds$}
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 $\operatorname{Re}\omega(k)$; multiplying by $e^{-\omega(k)t}$ changes the integrand to the causal factor $e^{-\omega(k)(t-s)}$.
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