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
!}}\int _{a}^{b}k(\theta )f^{(\nu +1)}(\theta )\,d\theta .}
Bounds
Several bounds on the value of follow from this result:
!}}\|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{aligned}}}
where , and are the taxicab, Euclidean and maximum norms respectively.[2]