Let $\mu$ be standard Gaussian measure on $\mathbb{C}^n$, i.e. $d\mu = \frac{1}{(2 \pi)^n} e^{-|z|^2/2}\,dz$, and fix $0 < p < 1$ (**note carefully**).



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> Suppose $g$ is holomorphic on $\mathbb{C}^n$ and satisfies $\int |g|^p\,d\mu < \infty$. Do there exist holomorphic polynomials $g\_n$ such that $\int |g-g\_n|^p\,d\mu \to 0$?
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Stated another way, let $\mathcal{H}$ be the space of holomorphic functions on $\mathbb{C}^n$ and $\mathcal{P}$ be the holomorphic polynomials. Let $\mathcal{H}L^p(\mu) = \{ g \in \mathcal{H} : \int |g|^p\,d\mu < \infty\}$, equipped with the usual metric $d(f,g) = \int |f-g|^p\,d\mu$ (which of course is not a norm). Clearly $\mathcal{P} \subset \mathcal{H} L^p(\mu)$. Is $\mathcal{P}$ dense in $\mathcal{H} L^p(\mu)$?


Even a proof or counterexample for $n=1$ would shed some light.


This is true for $1 \le p < \infty$ and seems to be well known, but the proof I've seen does not go through for $p < 1$, since it needs $t^p$ to be a convex function. It goes like this. First you show that as $\theta \to 0$ we have $g(e^{i\theta} \cdot) \to g$ in $L^p(\mu)$; this part still works for $p < 1$. Now let $F\_n$ be the [Fejér kernel](http://en.wikipedia.org/wiki/Fej%C3%A9r_kernel) and set $g\_n(z) = \int\_{-\pi}^\pi F\_n(\theta) g(e^{i \theta} z)\,d\theta$, which can be shown to be a polynomial of degree at most $n-1$. If $p \ge 1$ we can use Jensen's inequality on the probability measure $F(\theta) \,d\theta$ to show
$$\int\_{\mathbb{C}^n} |g(z)-g\_n(z)|^p\,\mu(dz) \le \int\_{\mathbb{C}^n} \int\_{-\pi}^\pi F\_n(\theta) \left|g(z)-g(e^{i\theta} z)\right|^p\,d\theta \,\mu(dz).$$
After interchanging the integrals, properties of the Fejér kernel imply this converges to $\lim\_{\theta \to 0} \int\_{\mathbb{C}^n} \left|g(z)-g(e^{i\theta} z)\right|^p\,d\mu = 0$ as previously argued. But for $p < 1$, Jensen's inequality goes the wrong way.


One idea is to try to find a dominating function for $|g-g\_n|$ and use dominated convergence. For instance, $g\_n$ is controlled by $\sup\_{\theta} |g(e^{i \theta} z)|$, and maybe we could bound that in terms of $g$ or some related $L^p$ function. But I don't see how to do that.



Are polynomials dense in holomorphic $L^p(\mathrm{Gauss})$ for $p < 1$?
Yes, they are dense.


The key result comes from a paper of R. Wallstén [1], in which Theorem 3.1 implies that the set $\mathcal{E}$ of functions of the form
$$f(z) = \sum\_{j=1}^m a\_j e^{\langle z, w\_j \rangle}, \qquad a\_j \in \mathbb{C}, \, w\_j \in \mathbb{C}^n$$
is dense in $\mathcal{H} L^p(\mu)$ for any $0 < p < 1$. Clearly $\mathcal{E} \subset \mathcal{H} L^1(\mu)$, and we already know that $\mathcal{P}$ is dense in $\mathcal{H} L^1(\mu)$. So given $g \in \mathcal{H} L^p(\mu)$ and $\epsilon > 0$, we may choose $f \in \mathcal{E}$ and $h \in \mathcal{P}$ with
$$\int |g-f|^p\,d\mu < \epsilon, \qquad \int |f-h|\,d\mu < \epsilon.$$
But by Jensen's inequality, $\int |f-h|^p \,d\mu \le \left(\int |f-h|\,d\mu\right)^p$, so by the triangle inequality in $L^p$ we have
$$\int |g-h|^p\,d\mu \le \int |g-f|^p\,d\mu + \int |f-h|^p \,d\mu \le \epsilon + \epsilon^p.$$


[1] Wallstén, Robert. 
The $S^p$-criterion for Hankel forms on the Fock space, $0<p<1$.
*Math. Scand.* **64** (1989), no. 1, 123–132. [MR 1036432](http://www.ams.org/mathscinet-getitem?mr=1036432) [Open access via DigiZeitschriften (PDF)](http://www.digizeitschriften.de/download/PPN35397434X_0064/log10.pdf)


