Is the EulerMclaurin sum formula valid for distributions ?by mhill Tags: distributions, eulermclaurin, formula, valid 

#1
Feb208, 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^{2r1}(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 (xc) [/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] floorx^2 + 3 [/tex] so its first derivative is just a Dirac delta. 


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