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HallsofIvy

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I'm not sure I can make sense out of your question. Over what interval are you taking the convolution? What happens if f(x) is a constant or f(x)= x?

It is true that the convolution, over [itex]-\infty[/itex] to [itex]\infty[/itex], of two Gaussians is a Gaussian so this may be a "fixed point" theorem.

It is true that the convolution, over [itex]-\infty[/itex] to [itex]\infty[/itex], of two Gaussians is a Gaussian so this may be a "fixed point" theorem.

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rasing any well defined function to a high integer power gives something increasingly similar to a Gaussian.

What makes you think this? In raising a function to a very high power, the parts with |f(x)|>1 get increasingly larger while those with |f(x)|<1 vanish. So, for example, if the function has several narrow peaks whose height is greater than one, its very high powers will have spikes where each of those peaks were and vanish everywhere else.

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HallsofIvy

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yep, that's what i mean

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StatusX

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so my original question was what types of functions will not end up with a gaussian?

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AlephZero

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so my original question was what types of functions will not end up with a gaussian?

Think about what convolution means in the frequency domain, and it's clear that almost any function will not end up as a gaussian.

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