# 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.

## Statement

Let ${\mathcal {V}}[a,b]$  be the space of all differentiable functions $f$  defined for $x\in (a,b)$  that are of bounded variation on $[a,b]$ , and let $L$  be a linear functional on ${\mathcal {V}}[a,b]$ . Assume that $f$  is ${\textstyle \nu +1}$  times continuously differentiable and that $L$  annihilates all polynomials of degree $\leq \nu$ , i.e.

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

Suppose further that for any bivariate function $g(x,\theta )$  with $g(x,\cdot ),\,g(\cdot ,\theta )\in C^{\nu +1}[a,b]$ , the following is valid:
$L\int _{a}^{b}g(x,\theta )\,d\theta =\int _{a}^{b}Lg(x,\theta )\,d\theta ,$

and define the Peano kernel of $L$  as
$k(\theta )=L[(x-\theta )_{+}^{\nu }],\qquad \theta \in [a,b],$

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

The Peano kernel theorem then states that
$Lf={\frac {1}{\nu !}}\int _{a}^{b}k(\theta )f^{(\nu +1)}(\theta )\,d\theta ,$

provided $k\in {\mathcal {V}}[a,b]$ .

### Bounds

Several bounds on the value of $Lf$  follow from this result:

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

## Application

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

{\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-a)^{\nu }f^{(\nu +1)}(\theta )\,d\theta ,\end{aligned}}

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