Peano kernel theorem explained

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

Let

l{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 l{V}[a,b] . Assume that that L annihilates all polynomials of degree \leq \nu , i.e.$$Lp=0,\backslash qquad\; \backslash forall\; p\backslash in\backslash mathbb\_\backslash nu[x].$$Suppose further that for any bivariate function g(x,\theta) with g(x, ⋅ ),g( ⋅ ,\theta)\in C ^{\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^[2] states that, if k\in l{V}[a,b] , then for every function f that is $\backslash nu+1$ times continuously differentiable, we have$$Lf=\backslash frac\backslash int\_a^bk(\backslash theta)f^(\backslash theta)\backslash ,d\backslash theta.$$

Bounds

Several bounds on the value of

Lf follow from this result:

where

\| ⋅ \| _{1} , \| ⋅ \| _{2} and \| ⋅ \| _{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 P _{\nu} . The theorem above follows from the Taylor polynomial for f with integral remainder:\begin{align} f(x) = f(a) + {} & (x - a)f'(a) + ((x - a) ^{2})/(2) f''(a) + … \\[ 6 pt] & … + ((x - a) ^{\nu})/(\nu!) f ^{(\nu)} (a) + (1)/(\nu!) \int ^{x(x - \theta)} _{a} ^{\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\in P _{\nu} to annihilate all but the final term on the right-hand side, and using the ( ⋅ ) _{+} notation to remove the x -dependence from the integral limits.^[3]

See also

• Divided differences

Notes and References

1. Book: Ridgway Scott, L.. Numerical analysis. limited. 2011. Princeton University Press. 9780691146867. Princeton, N.J.. 209. 679940621.
2. Book: Iserles, Arieh . A first course in the numerical analysis of differential equations . 2009 . Cambridge University Press . 9780521734905 . 2nd . Cambridge . 443–444 . 277275036 . limited.
3. Web site: Numerical Analysis. Iserles. Arieh. 1997. 2018-08-09.