In Wenland's book on Scattered Data Approximation, Chapter 3 and this paper, a seminorm is defined using multi-index notations:
$$
|u(x)|_{C^{m+1}(\Omega)} = \text{max}_{|\beta|=m+1} ||D^{\beta} u||_{L^{\infty}(\Omega)},
$$
for $u(x) \in C^{m+1}(\Omega)$, with the derivative being:
$$
D^{\alpha} f = \frac{\partial^{|\alpha|}}{\partial x_1^{\alpha_1} \cdots\partial x_n^{\alpha_n}}, \quad \alpha = (\alpha_1, \cdots, \alpha_n),
$$
for $|\alpha| = \alpha_1 + \alpha_2 + \cdots \alpha_n| \leq k$, $x \in \mathbb{R}^d$, $f \in C^k (\Omega)$.
I am having troubles understanding that semi-norm. Let's consider an example and say that with $\beta = 2$ and $\Omega \subset \mathbb{R}^d$ for $d=2$, then is it correct that you will have three second-order derivatives? Namely:
$$
\frac{\partial ^2}{\partial x^2}, \frac{\partial ^2}{\partial x \partial y}, \frac{\partial ^2}{\partial y^2}.
$$
Is this the right way of thinking about it? If that's the case, how should I think of the $L^\infty(\Omega)$ norm here? And then what does the maximum of $|\beta| = m+1$ do?
Best Answer
The notation here is that $\beta$ is not a number like $\beta=2$, but instead $\beta$ is a vector/list of indices $\beta = (1,1)$ whose sum $|\beta| = 1 + 1$ is $2$. So the constraint notation $|\beta|=2$ means to consider $(2,0), (1,1),$ and $(0,2)$. So you are correct that these correspond exactly to $\partial^2/\partial x^2, \partial^2/\partial x \partial y$, and $\partial^2/\partial y^2$. Notice here that you can use multi-index notation for derivatives to describe this: $D^{(0,2)}f$ means $\partial^2 f/\partial y^2$.
Thus, what the book is saying is that the value of the seminorm of $u$ in $C^{m+1}(\Omega)$ is the same as the max of the $L^\infty(\Omega)$ norms of its derivatives of order $m+1$.