A Unified Polynomial Approximation Scheme for Classical Function Spaces

We prove polynomial density theorems for a broad class of classical function spaces on domains
$\Omega \subset \IR^n$, in both local (compact-open) and global (Banach) topologies.
In the local regime we show that restrictions of polynomials are dense in $C^k(\Omega)$,
$C^\infty(\Omega)$, and $L^p_{loc}(\Omega), W^{k,p}_{loc}(\Omega)$ for arbitrary open $\Omega$.

In the global regime on bounded domains we re-prove the classical density results for
$C^0(\bar{\Omega})$ (Stone-Weierstrass) and $L^p(\Omega)$.
For higher-differentiable function spaces ($k \geq 1$), polynomial density depends on regularity of
the domain. We show density in $C^k_b(\bar{\Omega})$ for $C^k$-extension domains (e.g. smooth boundary)
and in $W^{k,p}(\Omega)$ for Sobolev-extension domains (e.g. Lipschitz boundary).

The proofs are organized around a single approximation pipeline: extension, compact truncation,
analytic regularization, and Taylor approximation. This yields straightforward, standard and unified
arguments that make explicit the domain assumptions underlying several folklore polynomial-density
statements; we also record counterexamples showing that such hypotheses are necessary for global
approximation in strong norms.

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