I've arrived at the following equation involving the convolution of two functions:(adsbygoogle = window.adsbygoogle || []).push({});

[itex]

f(x) = \int_{-\infty}^{\infty} f(t) g(t-x) dt = f(x) \ast g(x)

[/itex]

Where:

[itex]

g(z) = e^{-z^2/2}

[/itex]

In other words, a function convoluted with a Gaussian pdf results in the same function.

I've tried taking fourier transforms, realizing that the FT of a gaussian results in another Gaussian:

[itex]

F[f(x)] = F[f(x) \ast g(x)] = F[f(x)] \cdot F[g(x)]

[/itex]

But this results in the [itex] F[f(x)] [/itex] cancelling out, leaving me with just:

[itex]

1 = F[g(x)] = e^{-w^2/2}

[/itex]

Any suggestions?

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# Strange convolution equation

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