I came across a question asking how the proof that the transform $f(g) = g(x-a) + g(x+a)$ was Hermitian worked. I was a bit surprised that the proof was symbolic, because if you plot out the coefficients it's obvious.
By "plot out the coefficients" I mean "treat the point (x, y) as the flow from input=y to output=x". In the case of f$(g) = g(x-a) + g(x+a)$, the flows are from $x-a$ to $x$ and from $x+a$ to $x$. If we set $a$ to 2 w.l.o.g., then points matching $(x-2, x)$ and $(x+2, x)$ will have coefficient 1 while all other points will have coefficient 0. So we get two lines:
(The lines differing in color is an artifact of how I plotted the coefficients; they both represent coefficients equal to 1.)
This is just a continuous version of your typical finite matrix grid-of-numbers representation. Noting that the coefficients are not affected by transposing (mirroring about the x=-y axis), we see that the transform must be Hermitian.
The fact that the original proof was done symbolically made me wonder if it's common to plot the coefficients of linear transforms on $\mathbb{R}$ the way I'm doing. Are there gotchas? Are there better ways?
Other examples:
Phase (arg) of the Fourier transform's coefficients (contour plot):
LaPlace transform coefficients (contour plot):
