Homogeneous linear majorants of Oscillatory Functions: A Geometric and Asymptotic Study

Dinis Maria Domingos Pinto

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Authors:

Dinis Maria Domingos Pinto (Dinis Maria Domingos Pinto)

Published in

July 28, 2026

Abstract

This working paper presents a systematic geometric and asymptotic investigation into how linear functions bound periodically modulated growth, focusing on the prototype family $f_{a,k}(x) = x(1 + a sin(kx))$. By evaluating the asymptotic ratio $R(a)$ between the area under the minimal homogeneous linear majorant and the true geometric area beneath the curve as $T \to \infty$, this study uncovers several structural invariants: Scale Separation & Frequency Invariance: While frequency $k$ drastically alters local graph geometry on finite intervals, it vanishes entirely in the asymptotic limit, proving that long-term efficiency is purely amplitude-driven. Geometry of Sign Changes & Transcendental Optimization: Enforcing geometric area via absolute values $|f_{a,k}(x)|$ for $a > 1$ creates a smooth $C^1$ transition between non-negative and sign-changing regimes. The efficiency ratio achieves a unique global peak $R(a*) \approx 2.12533$ at a transcendental amplitude directly governed by the Dottie number ($y* = cos y*$). Transient Surface Dynamics: Analysis of the finite-horizon ratio $R(a,T)$ highlights short-term phase overshoots up to $R \approx 3.2536$ before long-term periodic averaging flattens the surface toward its asymptotic asymptote. Universal Averaging Principle: The framework is generalized beyond polynomial weights $x^n$ to arbitrary subexponential weights $w(x)$ satisfying $w'(x)/w(x) \to 0$. We formally prove that local asymptotic flatness (in the sense of Karamata) causes the weight to decouple from periodic oscillations, establishing that asymptotic majorant efficiency is universally governed by the ratio M/\mu between the peak envelope and the periodic mean. Keywords: Majorant Efficiency, Asymptotic Analysis, Karamata Theory, Dottie Number, Subexponential Weights, Periodic Integration, Oscillatory Functions.