Polynomial Eigenfunctions and Matrix Lyapunov Equations from Energy Balance Integrals

TL;DR

This paper develops a unified theoretical framework linking orthogonal polynomial systems and matrix Lyapunov equations via energy dissipation principles in stochastic systems, revealing their dual nature and mathematical foundations.

Contribution

It introduces a novel energy balance-based approach that unifies polynomial eigenfunctions and matrix Lyapunov equations, with rigorous mathematical derivations.

Findings

Orthogonal polynomials and Lyapunov equations are dual representations of energy dissipation.

Finite-dimensional projections of energy integrals reproduce classical matrix equations.

Adding uniform dissipation preserves polynomial eigenfunctions and ensures physical energy balance.

Abstract

We establish a unified theoretical framework that connects classical orthogonal polynomial systems to matrix Lyapunov equations through the fundamental physics of energy dissipation in stochastic dynamical systems. Starting from the energy balance principle in infinite-dimensional Hilbert spaces, we derive a master integral representation that naturally encompasses both spectral geometry and covariance dynamics. The theory reveals that established orthogonal polynomials (Zernike, Hermite, spherical harmonics) and matrix Lyapunov equations are dual manifestations of the same underlying energy dissipation structure. We provide rigorous mathematical foundations showing how finite-dimensional projections of infinite-dimensional energy integrals reproduce classical matrix equations, with specific structure determined by the symmetries of noise processes. The framework demonstrates that adding uniform dissipation to classical differential operators preserves their polynomial eigenfunction structure while ensuring the energy balance conditions required for physical consistency.