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]
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.
Suppose further that for any bivariate function
with
, the following is valid:
and define the Peano kernel of
as
using the notation
The Peano kernel theorem[1][2] states that, if
, then for every function
that is
times continuously differentiable, we have
Several bounds on the value of
follow from this result:
where
,
and
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
. 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://backend.710302.xyz/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]