I have a question about linear approximation in the multivariate case.\
Let $f:B^d_r\to \mathbb{R}$ be a real-valued $C^2$-function defined on the $d$-dimensional ball of radius $r$ centered at the origin. Assume further that $\frac{\partial ^2f(\mathbf{x})}{\partial x_i \partial x_j}<C$ for all $i,j\in\{1,2,3,...,d\}$ where $C$ is a constant.
I am interested in the max-error $\sup_{\mathbf{x}\in B^d_r}|f(\mathbf{x})-P_1(\mathbf{x})|$ where $P_1(x)$ is a first order multivariate polynomial.
I know from Taylors theorem we have
\begin{align} f(\mathbf{x})=f(\mathbf{0})+\sum_{i=1}^d\frac{\partial f}{\partial x_i}(\mathbf{0})x_i + \sum_{|\mathbf{\alpha}|=2}R_{\mathbf{\alpha}}(\mathbf{x})\mathbf{x}^{\alpha} \end{align}
where $\alpha$ is a multi-index and $|\mathbf{\alpha}|=\alpha_1+\alpha_2+...+\alpha_d$ and $\mathbf{x}^{\mathbf{\alpha}}=x_1^{\alpha_1}x_2^{\alpha_2}...x_d^{\alpha_d}$.
We can also bound $R_{\mathbf{\alpha}}(\mathbf{x})\leq \frac{1}{\mathbf{\alpha}!}C$.
Thus if we choose $P_1(\mathbf{x})=f(\mathbf{0})+\nabla f(\mathbf{0})\cdot \mathbf{x}$ then \begin{align} |f(\mathbf{x})-P_1(\mathbf{x})|\leq \sum_{|\alpha|=2}\frac{1}{\alpha !}C\mathbf{x}^{\alpha} \end{align} Im not sure how to proceed from here. Of course I can say that $\mathbf{x}^{\alpha}\leq r^2$ for each term since $|\alpha|=2$ and similarly $\frac{1}{\alpha!}\leq 1$ so I would get:
$|f(\mathbf{x})-P_1(\mathbf{x})|\leq \frac{(d+1)d}{2}Cr^2$ as there are exactly $\frac{(d+1)d}{2}$ multi-indices $\alpha$ with $|\alpha|=2$.
My questions:
- Can this bound be improved?
- Especially, is it necessary to get the dimension $d$ into the estimate?
Kind regards
