Marcinkiewicz--Zygmund Stability for Short Trigonometric Approximants in a Window Norm

We develop a classical analytic framework for a windowed L² energy functional associated with a smoothed and centered logarithmic derivative (ξ'/ξ)(σ+it). The argument is organized around a fixed window/norm setup and a decomposition into an atomic component, a localized near term, a tail term, and a smooth remainder, with all truncations and interchanges justified at the analytic level. The main outcome is a uniform near-to-polynomial reduction: on the window, the near term is represented by a short trigonometric polynomial Pt₀∈ PQ (with QΔ) up to an error that is negligible in the window norm.This converts the relevant energy estimates to finite-dimensional bounds for Pt₀ and links continuous and discrete L² control through Marcinkiewicz--Zygmund type sampling stability.