Best Ulam Constants for Higher-Order Linear Difference Equations in the Hyperbolic Case

Source abstract

For higher-order linear difference equations with constant coefficients on the nonnegative integers, the best Ulam constant is determined under the hyperbolicity assumption that no characteristic root lies on the unit circle. A Green-function estimate gives a Ulam constant in this regime. When the characteristic polynomial has at least two distinct roots and at least one root lies inside the unit circle, however, neither the existence of a best constant nor the sharpness of that estimate was known. The Green-function bound is sharp for every hyperbolic root configuration. The argument combines uniform exponential decay of bounded homogeneous corrections with a finitely supported perturbation whose phases are chosen to align the stable-past and unstable-future Green contributions at a distant index. Hence the best Ulam constant exists throughout the hyperbolic case and equals the $\ell^1\$-norm of the corresponding half-line Green kernel. This resolves the remaining open problem formulated by Yuan, Xu and Brzdęk (2025).