Let ${\displaystyle {\mathcal {V}}[a,b]}$  be the space of all functions ${\displaystyle f}$  that are differentiable on ${\displaystyle (a,b)}$  that are of bounded variation on ${\displaystyle [a,b]}$ , and let ${\displaystyle L}$  be a linear functional on ${\displaystyle {\mathcal {V}}[a,b]}$ . Assume that that ${\displaystyle L}$  annihilates all polynomials of degree ${\displaystyle \leq \nu }$ , i.e.

Bounds edit

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

where ${\displaystyle \|\cdot \|_{1}}$ , ${\displaystyle \|\cdot \|_{2}}$  and ${\displaystyle \|\cdot \|_{\infty }}$ are the taxicab, Euclidean and maximum norms respectively.^[2]

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

${\displaystyle {\begin{aligned}f(x)=f(a)+{}&(x-a)f'(a)+{\frac {(x-a)^{2}}{2}}f''(a)+\cdots \\[6pt]&\cdots +{\frac {(x-a)^{\nu }}{\nu !}}f^{(\nu )}(a)+{\frac {1}{\nu !}}\int _{a}^{x}(x-\theta )^{\nu }f^{(\nu +1)}(\theta )\,d\theta ,\end{aligned}}}$ 

defining ${\displaystyle L(f)}$  as the error of the approximation, using the linearity of ${\displaystyle L}$  together with exactness for ${\displaystyle f\in \mathbb {P} _{\nu }}$  to annihilate all but the final term on the right-hand side, and using the ${\displaystyle (\cdot )_{+}}$  notation to remove the ${\displaystyle x}$ -dependence from the integral limits.^[3]

• Divided differences