Mathematical theorem used in numerical analysis
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[edit]
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
as
using the notation
The
Peano kernel theorem[1][2] states that, if
, then for every function
that is
times
continuously differentiable, we have
Several bounds on the value of follow from this result:
where , and are the taxicab, Euclidean and maximum norms respectively.[2]
Application[edit]
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[edit]
References[edit]