Simultaneous polynomial approximation in Beurling-Sobolev spaces via Blaschke products
Assuming that $ϕ(t)=o(t^2)$ as $t\to0$, we establish a lemma on simultaneous polynomial approximation in Orlicz-Beurling-Sobolev spaces $\ell_a^ϕ$. These spaces, endowed with the Luxemburg norm $\Vert \cdot \Vert_{\ell^ϕ}$, generalize the classical Beurling-Sobolev spaces $\ell_a^p$ for $p>2$. More precisely, we prove that for every $\varepsilon>0$, every $v\in\mathbb{N}$ and every function $φ$ continuous on $\partial\mathbb{D}$, there exist a polynomial $P(z)=\sum_{k=v}^d a_k z^k$ and a compact set $K\subset\partial\mathbb{D}$ with $m(K)>1-\varepsilon$ such that \[\|P\|_{\ell^ϕ}\le\varepsilon \quad \text{and}\quad \|P-φ\|_K\le\varepsilon.\] The proof relies on a result of independent interest describing the asymptotic behaviour of the Luxemburg norm $\|B^k\|_{\ell^ϕ}$ of powers of a finite Blaschke product $B$ which is not a monomial. This behaviour is governed by the comparison between $ϕ(t)$ and $t^2$ near $0$: the norms remain bounded when $ϕ\asymp t^2$, tend to $0$ when $ϕ=o(t^2)$, and diverge to $+\infty$ when $t^2=o(ϕ(t))$. A key ingredient in the proof is the qualitative limit $\sup_{j\ge0}|\widehat{B^k}(j)|\to0$ as $k\to\infty$. As an application of the simultaneous approximation lemma, we derive the existence of functions in $\ell_a^ϕ$ with universal properties, including Menshov universality of Taylor partial sums and universality with respect to radial boundary limits.