The picture below shows (in red) a sum of Gaussian kernels with different means, but all having variance 1, i.e.
f(x)=12π−−√∑i=1ne−12(x−μi)2 .
. The green curves show what I get if I select a point x0, and then "fit" a scaled Gaussian distribution g to f about x0, in the sense that I determine c and μ0 in the function
g(x):=c2π−−√e−12(x−μ0)2
so that g(x0)=f(x0) and g′(x0)=f′(x0). From numerical experimentation, it seems to always be the case that g(x)≤f(x) for all x. Does anyone know why this is so?