Peano kernel theorem

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]

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]

• Divided differences