Peano kernel theorem

Statement

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 ${\displaystyle \le \nu }$, i.e.$${\displaystyle Lp=0,\mathrm{\forall }p\in {\mathbb{P}}_{\nu }[x].}$$Suppose further that for any bivariate function ${\displaystyle g(x,\theta )}$ with ${\displaystyle g(x,\cdot ),g(\cdot ,\theta )\in C^{\nu +1}[a,b]}$, the following is valid:$${\displaystyle L{\int }_a^{b}g(x,\theta )d\theta ={\int }_a^{b}Lg(x,\theta )d\theta ,}$$and define the Peano kernel of ${\displaystyle L}$ as$${\displaystyle k(\theta )=L[(x-\theta {)}_{+}^{\nu }],\theta \in [a,b],}$$using the notation$${\displaystyle (x-\theta {)}_{+}^{\nu }=\{\begin{array}{ll}(x-\theta {)}^{\nu },& x\ge \theta ,\\ 0,& x\le \theta .\end{array}}$$The Peano kernel theorem states that, if ${\displaystyle k\in \mathcal{V}[a,b]}$, then for every function ${\displaystyle f}$ that is $\nu +1$ times continuously differentiable, we have $${\displaystyle Lf=\frac{1}{\nu !}{\int }_a^{b}k(\theta )f^{(\nu +1)}(\theta )d\theta .}$$

Bounds

Several bounds on the value of ${\displaystyle Lf}$ follow from this result:$${\displaystyle \begin{array}{cc}|Lf|& \le \frac{1}{\nu !}\Vert k{\Vert }_1\Vert f^{(\nu +1)}{\Vert }_{\mathrm{\infty }}\\ |Lf|& \le \frac{1}{\nu !}\Vert k{\Vert }_{\mathrm{\infty }}\Vert f^{(\nu +1)}{\Vert }_1\\ |Lf|& \le \frac{1}{\nu !}\Vert k{\Vert }_2\Vert f^{(\nu +1)}{\Vert }_2\end{array}}$$

where ${\displaystyle \Vert \cdot {\Vert }_1}$, ${\displaystyle \Vert \cdot {\Vert }_2}$ and ${\displaystyle \Vert \cdot {\Vert }_{\mathrm{\infty }}}$are the taxicab, Euclidean and maximum norms respectively.