Peano kernel theorem

Let $\backslash mathcal\{V\}[a,b]$ be the space of all functions $f$ that are differentiable on $(a,b)$ that are of bounded variation on $[a,b]$, and let $L$ be a linear functional on $\backslash mathcal\{V\}[a,b]$. Assume that that $L$ annihilates all polynomials of degree $\backslash leq\; \backslash nu$, i.e.$$Lp=0,\backslash qquad\; \backslash forall\; p\backslash in\backslash mathbb\{P\}\_\backslash nu[x].$$Suppose further that for any bivariate function $g(x,\backslash theta)$ with $g(x,\backslash cdot),\backslash ,g(\backslash cdot,\backslash theta)\backslash in\; C^\{\backslash nu+1\}[a,b]$, the following is valid:$$L\backslash int\_a^bg(x,\backslash theta)\backslash ,d\backslash theta=\backslash int\_a^bLg(x,\backslash theta)\backslash ,d\backslash theta,$$and define the Peano kernel of $L$ as$$k(\backslash theta)=L[(x-\backslash theta)^\backslash nu\_+],\backslash qquad\backslash theta\backslash in[a,b],$$using the notationThe Peano kernel theorem{{Cite book |last=Iserles |first=Arieh |url=https://archive.org/details/firstcoursenumer00aise |title=A first course in the numerical analysis of differential equations |date=2009 |publisher=Cambridge University Press |isbn=9780521734905 |edition=2nd |location=Cambridge |pages=[https://archive.org/details/firstcoursenumer00aise/page/n464 443]–444 |oclc=277275036 |url-access=limited}} states that, if $k\backslash in\backslash mathcal\{V\}[a,b]$, then for every function $f$ that is $\backslash nu+1$ times continuously differentiable, we have

$$Lf=\backslash frac\{1\}\{\backslash nu!\}\backslash int\_a^bk(\backslash theta)f^\{(\backslash nu+1)\}(\backslash theta)\backslash ,d\backslash theta.$$

Several bounds on the value of $Lf$ follow from this result:$$\backslash begin\{align\}$$

|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{align}

where $\backslash |\backslash cdot\backslash |\_1$, $\backslash |\backslash cdot\backslash |\_2$ and $\backslash |\backslash cdot\backslash |\_\backslash infty$are the taxicab, Euclidean and maximum norms respectively.

In practice, the main application of the Peano kernel theorem is to bound the error of an approximation that is exact for all $f\backslash in\backslash mathbb\{P\}\_\backslash nu$. The theorem above follows from the Taylor polynomial for $f$ with integral remainder:

: $$

\begin{align}

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{align}

defining $L(f)$ as the error of the approximation, using the linearity of $L$ together with exactness for $f\backslash in\backslash mathbb\{P\}\_\backslash nu$ to annihilate all but the final term on the right-hand side, and using the $(\backslash cdot)\_+$ notation to remove the $x$-dependence from the integral limits.{{Cite web|url=http://www.damtp.cam.ac.uk/user/examples/D3Ll.pdf|title=Numerical Analysis|last=Iserles|first=Arieh|date=1997|access-date=2018-08-09}}

• Divided differences

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Category:Numerical analysis

Category:Theorems in mathematical analysis