 Centered second-derivative Peano kernel

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