For a project I'm working on, I was working with the following equation

$$ w(x) = \int k(x,y)v(y)dy $$

I noticed that if I choose

$$ k(x,y) = -\delta'(x-y) $$

Then we probably get (I haven't touched Dirac delta functions for years, please correct me if I am wrong.).

$$ v'(x) = -\int \delta'(x-y) v(y)dy \tag{1} $$

Where $\delta'$ is the derivative of the Dirac delta distribution as detailed here and $v'(x)$ is the derivative of $v(x)$.

My question is: Would it be possible to discretize (1) using some technique to get different finite difference schemes for derivatives? Presumably different definitions of $\delta$ using different limit of sequences of functions would lead to different finite difference schemes.

Something like

$$ v' = Pv $$

Where $P$ is a matrix, $v$ is a vector whose entries are values of $v(x)$ at equispaced locations and and $v'$ is a vector containing the finite difference approximation to the derivative.


1 Answer 1


You could, but why would you? We have systematic ways of obtaining finite difference stencils approximating derivatives that do not rely on the derivatives of functions that are zero everywhere and infinite in one place.


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