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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 $f(0)=f'(T)=0$, 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.

1

# RFC for definite integral connection to second derivative

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 $f(0)=f'(T)=0$, 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.
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