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]

Statement

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.

$${\displaystyle Lp=0,\qquad \forall p\in \mathbb {P} _{\nu }[x].}$$

${\displaystyle g(x,\theta )}$

${\displaystyle g(x,\cdot ),\,g(\cdot ,\theta )\in C^{\nu +1}[a,b]}$

$${\displaystyle L\int _{a}^{b}g(x,\theta )\,d\theta =\int _{a}^{b}Lg(x,\theta )\,d\theta ,}$$

${\displaystyle L}$

$${\displaystyle k(\theta )=L[(x-\theta )_{+}^{\nu }],\qquad \theta \in [a,b],}$$

$${\displaystyle (x-\theta )_{+}^{\nu }={\begin{cases}(x-\theta )^{\nu },&x\geq \theta ,\\0,&x\leq \theta .\end{cases))}$$

${\displaystyle k\in {\mathcal {V))[a,b]}$

${\displaystyle f}$

${\textstyle \nu +1}$

$${\displaystyle Lf={\frac {1}{\nu !))\int _{a}^{b}k(\theta )f^{(\nu +1)}(\theta )\,d\theta .}$$

Bounds

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]

Application

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]