# 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]

## Statement

Let ${\displaystyle {\mathcal {V}}[a,b]}$  be the space of all differentiable functions ${\displaystyle f}$  defined for ${\displaystyle x\in (a,b)}$  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 ${\displaystyle f}$  is ${\textstyle \nu +1}$  times continuously differentiable and 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
${\displaystyle k(\theta )=L[(x-\theta )_{+}^{\nu }],\qquad \theta \in [a,b],}$

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

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

provided ${\displaystyle k\in {\mathcal {V}}[a,b]}$ .[1][2]

### 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-a)^{\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]