Mathematical theorem used in numerical analysis
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[edit]
Let
be the space of all functions
that are differentiable on
that are of bounded variation on
, and let
be a linear functional on
. Assume that that
annihilates all polynomials of degree
, i.e.
![{\displaystyle Lp=0,\qquad \forall p\in \mathbb {P} _{\nu }[x].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1088ed1492b09db29a571202ae5a889c0aa1f66)
Suppose further that for any
bivariate function ![{\displaystyle g(x,\theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad353faedd3077be2daf2387d1dccef6a2b80972)
with
![{\displaystyle g(x,\cdot ),\,g(\cdot ,\theta )\in C^{\nu +1}[a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60f59d00ed362cd5d387eef0d55afbbd5f563d02)
, the following is valid:
![{\displaystyle L\int _{a}^{b}g(x,\theta )\,d\theta =\int _{a}^{b}Lg(x,\theta )\,d\theta ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f952cea37dd090b83563098d7e17305408e8f85f)
and define the
Peano kernel of
![{\displaystyle L}](https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8)
as
![{\displaystyle k(\theta )=L[(x-\theta )_{+}^{\nu }],\qquad \theta \in [a,b],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/608a1f43f7369fd3ad50ab9a72466a517e1c040e)
using the notation
![{\displaystyle (x-\theta )_{+}^{\nu }={\begin{cases}(x-\theta )^{\nu },&x\geq \theta ,\\0,&x\leq \theta .\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92c4eda4e8c602397deb636aac3811569028f80c)
The
Peano kernel theorem[1][2] states that, if
![{\displaystyle k\in {\mathcal {V}}[a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a64be74ceee3b987b20a49de1fa54912caf17e97)
, then for every function
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
that is
![{\textstyle \nu +1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa3f7fd3859f3d7370145e29f20123dfd31255fe)
times
continuously differentiable, we have
![{\displaystyle Lf={\frac {1}{\nu !}}\int _{a}^{b}k(\theta )f^{(\nu +1)}(\theta )\,d\theta .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/752ba387f98f03002a58ebefea29926e095f85fa)
Several bounds on the value of
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2764ece246dfc561f08d3443aa1a664a80db8e57)
where
,
and
are the taxicab, Euclidean and maximum norms respectively.[2]
Application[edit]
In practice, the main application of the Peano kernel theorem is to bound the error of an approximation that is exact for all
. The theorem above follows from the Taylor polynomial for
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-\theta )^{\nu }f^{(\nu +1)}(\theta )\,d\theta ,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/409615249d661a640a1ea889a6e9483501388925)
defining
as the error of the approximation, using the linearity of
together with exactness for
to annihilate all but the final term on the right-hand side, and using the
notation to remove the
-dependence from the integral limits.[3]
See also[edit]
References[edit]