@article{Moriya2026, 
author = {Netzer Moriya},
title = {Energy balancing integrals, orthogonal polynomial systems, and matrix Lyapunov equations},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {1},
pages = {2406-2429},
keywords = {energy balance, Lyapunov equations, Zernike polynomials, orthogonal polynomials, covariance operators, stochastic PDEs, Ornstein-Uhlenbeck processes},
url = {https://www.sciopen.com/article/10.3934/math.2026098},
doi = {10.3934/math.2026098},
abstract = {Starting from a steady-state energy–balance law for noisy, linearly damped fields, we derive an operator covariance equation in Lyapunov–Sylvester form. On the unit disk with self-adjoint boundary conditions, the corresponding generator is diagonalized by the Zernike polynomials, and finite-mode projection yields a matrix Lyapunov equation for modal covariances. We prove explicit a priori truncation-error bounds tailored to Zernike systems: in the diagonal (or diagonally dominant) case the operator-norm tail admits a closed-form expression and, for Kolmogorov-type spectra, decays at a rate        O    (      N          −      7              /            3        ); for general Hilbert–Schmidt noise covariances we obtain Hilbert–Schmidt tail bounds with explicit dependence on system parameters and dissipation rates. We further extend the formulation to exponentially correlated (Ornstein-Uhlenbeck) forcing via an augmented-state Lyapunov/Sylvester construction, yielding closed-form denominator shifts and a covariance-based inversion for the OU correlation time    τ. Numerical examples validate the Lyapunov solves, the derived error bounds, and the    τ recovery procedure.}
}