Peano kernel theorem
# 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 functions $f$ that are differentiable on $(a,b)$ that are of bounded variation on $[a,b]$, and let $L$ be a linear functional on ${\mathcal {V}}[a,b]$. Assume that 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

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

using the notation

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

The Peano kernel theorem states that, if $k\in {\mathcal {V}}[a,b]$, then for every function $f$ that is ν+1{\textstyle \nu +1} times continuously differentiable, we have

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

### 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:

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

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.

