Convolution of a Gaussian Distribution and a discontinuous function

*To*: mathgroup at smc.vnet.net*Subject*: [mg97011] Convolution of a Gaussian Distribution and a discontinuous function*From*: Dan629 <dangerrity at gmail.com>*Date*: Mon, 2 Mar 2009 05:20:53 -0500 (EST)

This is a repost of a reply on an earlier thread but I think it got buried so I'm putting it up as a new topic with a more relevant subject line. ------- I don't know if this is a forum where math questions are posted (as opposed to strictly Mathematica questions), but I'll ask anyway. The PDF of the circular uniform distribution in one dimension has two discontinuities at x = radius. This makes the required integration to do a convolution fail, and it fails in Fourier space as well. If I do a numeric convolution with ListConvolve, of course, I can easily get a result. Alternatively I can use a summation and get a symbolic result consisting of hundreds of summands. These work, but neither solution can be readily integrated or differentiated when the other independent variables are complicated as they are in my case. I'd really like to get a closed form symbolic formula for the convolution. Am I dreaming to think that Mathematica, with all it's rich complexity, can do something like this? Or it is simply a fact of math that a discontinuous function cannot be convolved with a Gaussian Distribution and no trick (DiracDelta, HeavisideTheta) or computational genius will make it work? Given a failed convolution, I can use the summation solution and then fit the resulting curve. I've used the Taylor and polynomial expansions along with FFTs, but I'm not able to get a close enough approximation. The convolution is a beautiful, continuous curve based on a sum of Gaussians and so intuitively it seems as though it would be easy to fit, but I can't seem to get it. It seems that an approximation of either the Gaussian Distribution or the circular uniform distribution would make the integration possible, too. But I struggle with getting suitable non-exponential approximations to either of those. So, for the illustration, look at this: vars = { ll -> .27, mm -> 0, ss -> .06 }; pdf1[ xx_ ] := PDF[ NormalDistribution[ mm, ss ], xx ]; pdf2[ xx_ ] := Piecewise[ { { 1/(Pi * Sqrt[ ll^2 - xx^2 ]), Abs[ xx ] < ll } } ]; Plot[ { pdf1[ xx ], pdf2[ xx ] } /. vars, { xx, -.3, .3 } ] conv = Sum[ (pdf1[ xx - zz ] pdf2[ zz ]), { zz, -.6, .6, 0.02 } ] / 60; Plot[ Evaluate[ conv /. vars ], { xx, -.5, .5 } ] I'm seeking a symbolic representation of the second curve. (All variables are real, and both ll and ss are always positive.) Please be kind Jens -- I'm admittedly an amateur!

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