= Centered second-derivative Peano kernel
{title2=$K(t)=\tfrac12(1+t)^2\ (t<0),\quad K(t)=-\tfrac12(1-t)^2\ (t>0)$}
For $L(f)=f''(0)-f(-1)+2f(0)-f(1)$ on $C^3[-1,1]$, quadratic polynomials are annihilated. The <Peano kernel theorem> gives $L(f)=\int_{-1}^1K(t)f'''(t)\,dt$, with the displayed kernel; its value at zero does not affect the integral. Its absolute integral is $1/3$, proving $|L(f)|\le\|f'''\|_\infty/3$. The kernel changes sign, consistent with exactness also on cubic polynomials despite the estimate using only three derivatives.
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