Direct smoothing with logarithmic distances doesn't conserve the norm.
The direct smoothing doesn't conserve the norm of an input field and also distributes the weight differently depending on the scale.
from nifty import *
x = RGSpace((2048), harmonic=True)
xp = PowerSpace(x)
f = Field(xp, val = 1.)
f.val[100] = 10
f.val[500] = 10
f.val[50] = 10
sm = SmoothingOperator(xp, sigma=0.1, log_distances=True)
g = sm(f)
f.dot(f)
g.dot(g)
plo = plotting.PowerPlotter(title='power.html')
plo(g)