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Is the Euler-Mclaurin sum formula valid for distributions ?

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mhill
#1
Feb2-08, 05:16 AM
P: 193
If we take the Euler MacLaurin sum formula

[tex] \int_{a}^{b} dxf(x) - \sum_{n=a} ^{b}f(n)= (1/2)(f(b)+f(a))+ \sum_{r}B_{2r}(2n!)^{-1}D^{2r-1}(f(b)-f(a)) [/tex]

However let's suppose that f(x) or the first , third , ... derivative is just for example a Dirac delta distribution or [tex] D^{k}\delta (x-c) [/tex]

We could use distribution theory to justify the derivative of a Dirac delta, so the Euler Mac Laurin sum formula would yied to distribution instead of real valued function but i am not sure.

to put an example, let suppose that f(x) is proportional to the floor function [tex] floor|x^2 + 3| [/tex] so its first derivative is just a Dirac delta.
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