Peano kernel

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 \mathcal ,b/math> be the space of all functions $f$ that are differentiable on $(a,b)$ that are of bounded variation on ,b/math>, and let $L$ be a linear functional on \mathcal ,b/math>. Assume that that $L$ ''annihilates'' all polynomials of degree $\leq \nu$, i.e.Lp=0,\qquad \forall p\in\mathbb_\nu Suppose further that for any bivariate function $g(x,\theta)$ with g(x,\cdot),\,g(\cdot,\theta)\in C^ ,b/math>, the following is valid:$L\int_a^bg(x,\theta)\,d\theta=\int_a^bLg(x,\theta)\,d\theta,$and define the Peano kernel of $L$ ask(\theta)=L x-\theta)^\nu_+\qquad\theta\in ,busing the notation$(x-\theta)^\nu_+ = \begin (x-\theta)^\nu, & x\geq\theta, \\ 0, & x\leq\theta. \end$The ''Peano kernel theorem'' states that, if k\in\mathcal ,b/math>, then for every function $f$ that is $\nu+1$ times continuously differentiable, we have
$Lf=\frac\int_a^bk(\theta)f^(\theta)\,d\theta.$

Bounds

Several bounds on the value of $Lf$ follow from this result:\begin
, Lf, &\leq\frac\, k\, _1\, f^\, _\infty\\ pt, Lf, &\leq\frac\, k\, _\infty\, f^\, _1\\ pt, Lf, &\leq\frac\, k\, _2\, f^\, _2
\end

where $\, \cdot\, _1$, $\, \cdot\, _2$ and $\, \cdot\, _\infty$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 $f\in\mathbb_\nu$. The theorem above follows from the Taylor polynomial for $f$ with integral remainder:

:
\begin
f(x)=f(a) + & (x-a)f'(a) + \fracf''(a)+\cdots \\ pt& \cdots+\fracf^\nu(a)+
\frac\int_a^x(x-\theta)^\nu f^(\theta)\,d\theta,
\end

defining $L(f)$ as the error of the approximation, using the linearity of $L$ together with exactness for $f\in\mathbb_\nu$ to annihilate all but the final term on the right-hand side, and using the $(\cdot)_+$ notation to remove the $x$-dependence from the integral limits.

See also

* Divided differences

References

{{reflist

Numerical analysis