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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 ${\displaystyle {\mathcal {V}}}$ be the space of all functions ${\displaystyle f}$ that are differentiable on ${\displaystyle (a,b)}$ that are of bounded variation on ${\displaystyle }$, and let ${\displaystyle L}$ be a linear functional on ${\displaystyle {\mathcal {V}}}$. Assume that that ${\displaystyle L}$ annihilates all polynomials of degree ${\displaystyle \leq \nu }$, i.e.$${\displaystyle Lp=0,\qquad \forall p\in \mathbb {P} _{\nu }.}$$Suppose further that for any bivariate function ${\displaystyle g(x,\theta )}$ with ${\displaystyle g(x,\cdot ),\,g(\cdot ,\theta )\in C^{\nu +1}}$, 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$${\displaystyle k(\theta )=L,\qquad \theta \in ,}$$using the notation$${\displaystyle (x-\theta )_{+}^{\nu }={\begin{cases}(x-\theta )^{\nu },&x\geq \theta ,\\0,&x\leq \theta .\end{cases}}}$$The Peano kernel theorem states that, if ${\displaystyle k\in {\mathcal {V}}}$, then for every function ${\displaystyle f}$ that is ${\textstyle \nu +1}$ times continuously differentiable, we have $${\displaystyle Lf={\frac {1}{\nu !}}\int _{a}^{b}k(\theta )f^{(\nu +1)}(\theta )\,d\theta .}$$

Bounds

Several bounds on the value of ${\displaystyle Lf}$ follow from this result:$${\displaystyle {\begin{aligned}|Lf|&\leq {\frac {1}{\nu !}}\|k\|_{1}\|f^{(\nu +1)}\|_{\infty }\\|Lf|&\leq {\frac {1}{\nu !}}\|k\|_{\infty }\|f^{(\nu +1)}\|_{1}\\|Lf|&\leq {\frac {1}{\nu !}}\|k\|_{2}\|f^{(\nu +1)}\|_{2}\end{aligned}}}$$

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

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:

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

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.

See also

• Divided differences

References

1. ^ Ridgway Scott, L. (2011). Numerical analysis. Princeton, N.J.: Princeton University Press. pp. 209. ISBN 9780691146867. OCLC 679940621.
2. ^ Iserles, Arieh (2009). A first course in the numerical analysis of differential equations (2nd ed.). Cambridge: Cambridge University Press. pp. 443–444. ISBN 9780521734905. OCLC 277275036.
3. Iserles, Arieh (1997). "Numerical Analysis" (PDF). Retrieved 2018-08-09.

• Numerical analysis