Can we approximate this function by a polynomial?

Question

For the more mathematically minded,

we have $x \in \mathbb{R}^2$ and the function $h(x)$ defined as:

$h(x)=\alpha_1x_1^2+\alpha_2x_2^2+\alpha_3x_1+\alpha_4x_2+\alpha_5x_1x_2+\alpha_6$

and the vector of alpha's is known and further guaranteed to be such that $h(x)$ is a general elliptic paraboloid (i.e. a convex function).

Now, define $g(x)=(h(x))^+$ where $(z)^+$ is the positive part of $z\in\mathbb{R}$.

The questions follows:

1. Is there a way to globally approximate $g(x)$ by a polynomial ? If so what it is?
2. Can this approach (the answer to the previous sub-question) be extended to $x\in\mathbb{R}^p$ with $p$ moderatly large ?
3. Would the problem be any easier if we were to assume $\alpha_5=0$ ?

Following Whuber's comment: is it possible to find a polynomial approximation to $g(x)$ that would be better than $h(x)$? in the event that many approximations solution for this problem exist, what are they ?

I can obviously solve $argmin.\int_{\mathbb{R}^2}(g(x)-\hat{g}(x))^2 dx$. I'm wondering if there are explicit, known solutions to this problem (i.e. polynomial series for example of which i know close to nothing).

Answer 1

The $L^2$ distance between $g$ and a polynomial approximation will be finite if and only if the polynomial approximation behaves asymptotically like $g$, which means it behaves asymptotically like $h$, which implies it must equal $h$. Therefore $h$ is the unique $L^2$ approximator.

Thus:

(1) Yes; the global approximator to $g$ is $h$.

(2) Yes; the same reasoning holds for all finite $p$.

(3) No; the value of $\alpha_5$ makes no difference.

Follow-up question (i): No, you cannot do better than $h$.

Follow-up question (ii) [how many approximations]: Not applicable due to the answer to (i).

You can get a considerably better set of answers if you're willing to limit the domain of $g$ to a compact [measurable] subset.

Answer 2

Note that $h(x)$ itself is a polynomial in $x_1$ and $x_2$ and $g(x)$ is what I would call a truncated polynomial.

The region where $h(x)$ is negative, $g(x)$ will be zero. You can approximate this region where $h(x)$ is flat by a non-constant polynomial (non-constant because you also need to approximate the region where $h(x) > 0$) but this will always be a 'wiggly' approximation (at least for a finite number of terms in the approximating polynomial).

It's not clear to me what you want to achieve, $g(x)$ actually has a quite simple analytical form and maybe for your problem you just have to consider the two regions $h(x) < 0$ and $h(x) \ge 0$ separately.