= Real bounded Fourier modes of the Klein-Gordon equation
With inverse-time convention $e^{i\omega t}$, the half-line <Klein-Gordon equation> $\sigma_{tt}-\sigma_{rr}+\mu^2\sigma=0$ has spatial <normal modes> $Ae^{ikr}+Be^{-ikr}$, where $k=\sqrt{\omega^2-\mu^2}>0$ above threshold and $k=i\sqrt{\mu^2-\omega^2}$ below it. For the branch even in $\omega$, reality above threshold requires $A(\omega)=B(-\omega)^*$. Below threshold boundedness eliminates $B$, and reality requires $A(\omega)=A(-\omega)^*$. At threshold the spatial <differential equation> instead has the <basis> $1,r$; boundedness retains only the constant mode. A spectral <integral> with ordinary <integrable functions> as amplitudes ignores the isolated threshold points, while <distribution (mathematical analysis)> amplitudes can represent the surviving monochromatic constant-spatial modes.
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