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.
be the space of all functions
that are differentiable on
that are of bounded variation on
be a linear functional on
Assume that that
annihilates all polynomials of degree
Suppose further that for any bivariate function
, 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 (x-\theta )_{+}^{\nu }={\begin{cases}(x-\theta )^{\nu },&x\geq \theta ,\\0,&x\leq \theta .\end{cases}}}
The Peano kernel theorem[1][2] states that, if
, then for every function
times continuously differentiable, we have
Several bounds on the value of
{\displaystyle Lf}
follow from this result:
{\displaystyle {\begin{aligned}|Lf|&\leq {\frac {1}{\nu !
}}\|k\|_{2}\|f^{(\nu +1)}\|_{2}\end{aligned}}}
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: 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.