I have a very simple question, because I somehow don’t get this definition. The version of the Peano kernel theorem that we had in class is:
For all linear functionals $\mathcal{L}:C^{n+1}([a,b]) \rightarrow \mathbb{R}$ of the form:
$$ \mathcal{L}(f)=\int_a^b (a_0(x)f(x)+a_1(x)f’(x)+\dots+a_n(x)f^{(n)}(x))dx+\sum_{i=1}^{j_0} b_{i,0}f(x_{i,0})+ \sum_{i=1}^{j_1} b_{i,1}f’(x_{i,1})+\dots + \sum_{i=1}^{j_n} b_{i,n}f^{(n)}(x_{i,n}) $$ (the $a_i(x)$ piecewise continuous)
we have: $\mathbb{P}_n \subseteq \ker(\mathcal{L}) \Rightarrow \mathcal{L}(f)=\int_a^b f^{(n+1)}(t)K(t)dt$, ($\mathbb{P}_n$ is the space of polynomials with degree at most $n$) with the peano kernel:
$$ K(t):= \frac{1}{n!}\mathcal{L}((x-t)^n_+), \quad (x-t)^n_+ := \begin{cases} (x-t)^n, \quad x \geq t \\ 0, \quad \quad \ \ \quad x<t \end{cases} $$
Now my questions is “Why is $K(t)$ (and thus $\mathcal{L}$) not always $0$, since $(x-t)^n \in \mathbb{P}_n$“.
If someone knows this theorem I’d be very happy if he/she could clear up my confusion:)