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Peano kernel theorem

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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

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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 asusing the notationThe Peano kernel theorem[1][2] states that, if , then for every function that is times continuously differentiable, we have

Bounds

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Several bounds on the value of follow from this result:

where , and are the taxicab, Euclidean and maximum norms respectively.[2]

Application

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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.[3]

See also

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References

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  1. ^ a b Ridgway Scott, L. (2011). Numerical analysis. Princeton, N.J.: Princeton University Press. pp. 209. ISBN 9780691146867. OCLC 679940621.
  2. ^ a b Iserles, Arieh (2009). A first course in the numerical analysis of differential equations (2nd ed.). Cambridge: Cambridge University Press. pp. 443–444. ISBN 9780521734905. OCLC 277275036.
  3. ^ Iserles, Arieh (1997). "Numerical Analysis" (PDF). Retrieved 2018-08-09.