Research seminar in Mathematics by Johan Andersson

Abstract

We report on our current research project where we prove the following results:

Let $\varphi: [0,\infty) \to [0,\infty)$ be a strictly increasing continuous function with $\varphi(0)=0$.

Then there exists a non-trivial entire function $f$ such that

\begin{gather*}

\int_{\mathbb{C}} \varphi(|f(z)|) dA(z) < \infty  \\

\intertext{if and only if}

\int_0^\infty \frac{dx}{1+\varphi(e^{e^x})} = \infty.

\end{gather*}

Furthermore, when this divergence condition holds, the space of entire functions is dense in the Orlicz space $L_\varphi(\mathbb{C})$.

Similarly we prove that for any compact set $K \subset \mathbb{C}$ with non-empty interior, then the set of polynomials is  dense in $L_\varphi(K)$ if and only if this integral diverges.

Remarkably, this allows us to approximate non-analytic functions, such as $f(z)=\bar{z}$, by polynomials in this Orlicz space.

In particular, our results resolves a gap left open by Kalton (1980) for the case where $\varphi(x) = (\log^+ \log^+ x)^p$ with $1 < p \le 2$.  

Methods of proofs include a Carleman-type differential inequality and the tangential Arakelyan approximation theorem.  

Note: This is our first piece of research where we have used a large language model (Gemini Pro 2.5, 3.0, 3.1) as an assistant, helpful for brain storming, editing part of the text and finding relevant references.