arXiv:2611.13288v1 Announce Type: new Abstract: We give a counterexample to a question of Borichev on polynomial approximation with monotone weights. We construct an even continuous weight $W$, strictly increasing on $[0,\infty)$, with superpolynomial growth and finite logarithmic integral, and an entire function $F$ of zero exponential type such that $F/W$ vanishes at infinity and belongs to $L^p(\mathbb{R})$ for every $1\leq p<\infty$. Nevertheless, $F$ has positive distance from the polynomials in the weighted uniform norm and in every corresponding weighted $L^{p}$ norm.
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