Support classes

The classification witnesses have literal compact smooth Cauchy data. The next section uses Schwartz data with a Fourier gap. These conditions must not be identified.

A compactly supported smooth datum has an entire Fourier transform. If that transform vanishes on a nonempty real open set, real-analytic uniqueness forces the datum to vanish. Thus a nonzero Fourier-gap Schwartz wave cannot also have compact Cauchy data.

Nor does compact spacetime support force an ungauged vacuum Einstein perturbation to vanish: compact pure gauges provide counterexamples. The corresponding uniqueness statement is valid for a homogeneous normally hyperbolic wave equation with zero data before the support.

Figure fig:support-and-restriction separates the support classes and the restricted extension problem from the full-space classification.