Hi,


During my research I found an interesting fact, and I'd like to know if it's interesting for others as well.
Find a function $g(x,t):[0,T]\times[0,T]\rightarrow[0,T]$ such that for any twice differentiable $f(x):[0,T]\rightarrow[0,T]$ such that $f(0)=f'(0)=0$, the equality
$$ f(x)=\intop\_0^Tf''(t)g(x,t)dt$$
holds. Note that $g$ is independent of $f$.


I found such a $g$, and I'll post it as an answer soon. I'd like to know if this is simple/known/interesting.



RFC for definite integral connection to second derivative
this looks like a simple consequence, upon twice partial integration, of $f(x)=\int\_0^T f(t)\delta(t-x)dt$, so your $g(x,t)=(x-t)\theta(x-t)$


$\delta(x)=d\theta(x)/dx$ relates Dirac delta function and Heaviside step function.


