Peano kernel theorem

In numerical analysis, the Peano kernel theorem is a general result on error bounds for a wide class of numerical approximations (such as numerical quadratures), defined in terms of linear functionals. It is attributed to Giuseppe Peano.¹

Let ${\displaystyle {\mathcal {V}}[a,b]}$ be the space of all functions ${\displaystyle f}$ that are differentiable on ${\displaystyle (a,b)}$ that are of bounded variation on ${\displaystyle [a,b]}$, and let ${\displaystyle L}$ be a linear functional on ${\displaystyle {\mathcal {V}}[a,b]}$. Assume that that ${\displaystyle L}$ annihilates all polynomials of degree ${\displaystyle \leq \nu }$, i.e.$${\displaystyle Lp=0,\qquad \forall p\in \mathbb {P} _{\nu }[x].}$$Suppose further that for any bivariate function ${\displaystyle g(x,\theta )}$ with ${\displaystyle g(x,\cdot ),\,g(\cdot ,\theta )\in C^{\nu +1}[a,b]}$, the following is valid:$${\displaystyle L\int _{a}^{b}g(x,\theta )\,d\theta =\int _{a}^{b}Lg(x,\theta )\,d\theta ,}$$and define the Peano kernel of ${\displaystyle L}$ as$${\displaystyle k(\theta )=L[(x-\theta )_{+}^{\nu }],\qquad \theta \in [a,b],}$$using the notation$${\displaystyle (x-\theta )_{+}^{\nu }={\begin{cases}(x-\theta )^{\nu },&x\geq \theta ,\\0,&x\leq \theta .\end{cases}}}$$The Peano kernel theorem¹² states that, if ${\displaystyle k\in {\mathcal {V}}[a,b]}$, then for every function ${\displaystyle f}$ that is ${\textstyle \nu +1}$ times continuously differentiable, we have $${\displaystyle Lf={\frac {1}{\nu !}}\int _{a}^{b}k(\theta )f^{(\nu +1)}(\theta )\,d\theta .}$$

Several bounds on the value of ${\displaystyle Lf}$ follow from this result:$${\displaystyle {\begin{aligned}|Lf|&\leq {\frac {1}{\nu !}}\|k\|_{1}\|f^{(\nu +1)}\|_{\infty }\\[5pt]|Lf|&\leq {\frac {1}{\nu !}}\|k\|_{\infty }\|f^{(\nu +1)}\|_{1}\\[5pt]|Lf|&\leq {\frac {1}{\nu !}}\|k\|_{2}\|f^{(\nu +1)}\|_{2}\end{aligned}}}$$

where ${\displaystyle \|\cdot \|_{1}}$, ${\displaystyle \|\cdot \|_{2}}$ and ${\displaystyle \|\cdot \|_{\infty }}$are the taxicab, Euclidean and maximum norms respectively.²

In practice, the main application of the Peano kernel theorem is to bound the error of an approximation that is exact for all ${\displaystyle f\in \mathbb {P} _{\nu }}$. The theorem above follows from the Taylor polynomial for ${\displaystyle f}$ with integral remainder:

${\displaystyle {\begin{aligned}f(x)=f(a)+{}&(x-a)f'(a)+{\frac {(x-a)^{2}}{2}}f''(a)+\cdots \\[6pt]&\cdots +{\frac {(x-a)^{\nu }}{\nu !}}f^{(\nu )}(a)+{\frac {1}{\nu !}}\int _{a}^{x}(x-\theta )^{\nu }f^{(\nu +1)}(\theta )\,d\theta ,\end{aligned}}}$

defining ${\displaystyle L(f)}$ as the error of the approximation, using the linearity of ${\displaystyle L}$ together with exactness for ${\displaystyle f\in \mathbb {P} _{\nu }}$ to annihilate all but the final term on the right-hand side, and using the ${\displaystyle (\cdot )_{+}}$ notation to remove the ${\displaystyle x}$-dependence from the integral limits.³