Shifted tapered-pole approximation on finite sectors: an exact asymptotic and dimension control

We study explicit lightning-plus-polynomial (LP) approximation of $g(z)z^{\alpha }(\log z{)}^m$ on a closed unit-radius sector, where $0<\alpha <1$, $m$ is fixed, and $g$ is analytic nearby. Since the basic sector root-exponential rate is already known, we address three finer questions. First, a one-sided Abel–Poisson formula and a pole-free notched contour give a branchwise error bound that is uniform for grid shifts in compact subsets of $(0,1)$ and valid for every fixed logarithmic power. Second, for $z^{\alpha }$ in the dense and balanced regimes, an exact vertex-scale rescaling identifies a positive finite leading constant for a specified shifted sequence. Third, a rate-matched band of additional representation poles followed by polynomial compression gives the explicit representation dimension $N+2\sqrt{2\alpha {\lambda }_{\sigma }N}+O(1)$ while leaving the first $N$ tapered poles unchanged. A slightly overresolved hybrid sequence also preserves the exact leading constant. Matched-dimension experiments, including reentrant sectors, illustrate the allocation. These exact-asymptotic and dimension statements concern the stated constructions, not sector minimax approximation or total computational complexity.