---
tag: 7N8E1BR67D
kind: statement
role: theorem
name: Peano kernel theorem
msc: [41A80, 41A55, 65D32]
uses: [ADSVMCY77A, 0HFSFVMWQB, 0WJGR81HBD]
about: [ADSVMCY77A]
sources:
  - url: https://en.wikipedia.org/wiki/Peano_kernel_theorem
same_as:
  wikidata: Q60741336
  wikipedia: Peano kernel theorem
---

## Statement

Let $a < b$ and $k \ge 0$ be given, and write $(t)_+^k$ for $t^k$ when $t \ge 0$ and $0$ when $t < 0$.
Let $\alpha_0, \dots, \alpha_k$ be piecewise continuous functions on $[a, b]$, and let $m_0, \dots, m_k \ge 0$ be integers.
For $0 \le j \le k$ and $1 \le i \le m_j$, let $\beta_{ij}$ be a real number and $x_{ij}$ a point of $[a, b]$.
Let $L$ be the linear functional on $C^{k+1}[a, b]$ given by
$$L(f) = \sum_{j=0}^{k} \Big( \int_a^b \alpha_j(x) f^{(j)}(x)\,dx + \sum_{i=1}^{m_j} \beta_{ij} f^{(j)}(x_{ij}) \Big).$$
Assume that $L(p) = 0$ for every polynomial $p$ of degree at most $k$.
Then for every $f \in C^{k+1}[a, b]$,
$$L(f) = \int_a^b K(\theta)\, f^{(k+1)}(\theta)\,d\theta, \qquad K(\theta) = \frac{1}{k!}\, L_x\big[(x - \theta)_+^k\big],$$
where $L_x$ means that $L$ acts on $(x - \theta)_+^k$ as a function of $x$ for fixed $\theta$.
The function $K$ is the *Peano kernel* of $L$.

## Remarks

Proof idea: write $f$ by [[Taylor's theorem with the integral remainder|0WJGR81HBD]] at $a$, as the [[Taylor polynomial|0HFSFVMWQB]] of degree $k$ plus $\frac{1}{k!} \int_a^b (x - \theta)_+^k f^{(k+1)}(\theta)\,d\theta$.
The functional kills the polynomial part.
The special form of $L$ allows $L$ to be moved inside the integral over $\theta$, which gives the formula.

If $K$ does not change sign on $[a, b]$, the mean value theorem for integrals gives $L(f) = \frac{f^{(k+1)}(\xi)}{(k+1)!}\, L\big(x^{k+1}\big)$ for some $\xi \in [a, b]$.
Also $\abs{L(f)} \le \norm{K}_{L^1} \, \norm{f^{(k+1)}}_\infty$ in all cases.

Example: the error of the [[trapezoidal rule|FH0538XCZR]], $L(f) = \int_a^b f - \frac{b - a}{2} \big( f(a) + f(b) \big)$, with $k = 1$.
Here $K(\theta) = \int_\theta^b (x - \theta)\,dx - \frac{b - a}{2} (b - \theta) = -\frac12 (\theta - a)(b - \theta)$, which is $\le 0$.
Since $\int_a^b K = -\frac{(b - a)^3}{12}$, one gets $L(f) = -\frac{(b - a)^3}{12} f''(\xi)$ for some $\xi$.

The hypothesis $L(p) = 0$ for degree at most $k$ is needed: for $L(f) = f(0)$ on $[0, 1]$ and $k = 0$, the formula would give $f(0) = \int_0^1 K f'$, which fails for $f = 1$.

## Explanations

### L1 human

Many approximation formulas, such as quadrature rules, are exact for all polynomials up to some degree.
The theorem says that their error is then an average of a high derivative of the function, with a fixed weight called the Peano kernel.
So the error is small when that derivative is small.
The kernel depends only on the formula, not on the function.

### L2 human

The theorem turns the question "how large is the error of this rule" into a computation with one explicit function $K$.
It is the standard way to derive the error terms of Newton–Cotes rules, of Gaussian rules and of interpolation and differentiation formulas.
When $K$ keeps one sign, the error has the familiar form "constant times $f^{(k+1)}(\xi)$", and the constant is found by testing the rule on $x^{k+1}$.
When $K$ changes sign, one still has the bound by $\norm{K}_{L^1} \norm{f^{(k+1)}}_\infty$, which is sharp over the class of functions with bounded $(k+1)$-st derivative.
The [[degree of exactness|ZYT1JNT1Y0]] of a rule fixes the largest $k$ for which the theorem applies.