 Peano kernel theorem

The Peano kernel theorem converts polynomial exactness into a derivative-based error formula. Let $r\ge 1$ and let $L$ be a linear error functional annihilating every polynomial of degree below $r$. A useful sufficient setting is a finite combination of point values, derivatives of orders below $r$, and integrals against finite weights. For $f\in C^r[a,b]$,Here $(x-t{)}_{+}^{r-1}$ means $(x-t{)}^{r-1}$ for $x>t$ and zero for $x<t$; values at its finitely many evaluation breakpoints can be chosen arbitrarily for the outer integral. The proof applies $L$ to Taylor's theorem with integral remainder: the polynomial part vanishes, and the permitted evaluations and integrations interchange with the remainder integral. For a more general linear functional, that interchange is an additional hypothesis. The same construction makes B-splines Peano kernels of divided differences.^[ref]

For the midpoint error $L(f)={\int }_a^{b}f-(b-a)f((a+b)/2)$, use $r=2$. Its nonnegative kernel equals $(t-a{)}^2/2$ on the left half and $(b-t{)}^2/2$ on the right half. Its integral is $(b-a{)}^3/24$, immediately yielding the sharp quadrature error bound.

The Peano kernel theorem is an important result in the field of real analysis, particularly in the context of approximation theory and integral equations. Named after the Italian mathematician Giuseppe Peano, it deals with the approximation of continuous functions using integral operators.