Fractional Green-Operator Modeling of Optical Luminance Transfer: A Review Study on Exact Norms, Stability, and Certified Nystrom Approximation

This work develops a rigorous fractional Green-operator framework for one-dimensional optical luminance transfer with spatially nonlocal effects. The physical point-spread operator, which maps an internal luminance state to the measured image-plane response, is distinguished explicitly from the Green operator associated with a Riemann–Liouville fractional Dirichlet problem of order [Formula: see text]. The corresponding Green kernel is derived, proved to be continuous and nonnegative, and shown to preserve physically admissible nonnegative luminance states. Its exact operator norm on [Formula: see text] is computed as [Formula: see text] This exact constant provides a unified and directly verifiable basis for the qualitative analysis. Existence is established under sublinear growth without requiring a contraction hypothesis, while uniqueness follows from the explicit condition [Formula: see text]. Quantitative estimates are subsequently derived for positivity, solution magnitude, continuous dependence on the nonlinear optical response, propagation of perturbations through the measurement operator, Ulam–Hyers and generalized Ulam stability, and a priori and a posteriori convergence of Picard iteration. A Gauss–Legendre Nyström scheme is formulated, and the total approximation error is decomposed into quadrature-consistency and nonlinear-iteration contributions. Table-based computations demonstrate systematic mesh convergence, with the reference discrepancy decreasing from [Formula: see text] to [Formula: see text]. They also show that increasing the fractional order from [Formula: see text] to [Formula: see text] enlarges the certified stability margin and reduces both peak and integrated luminance for the considered benchmark. Structured forcing perturbations exhibit linear response amplification well below the rigorous theoretical bound. These results connect exact fractional-operator analysis with stable and reproducible computation for nonlocal optical-transfer models.