Global Conference on Mathematics and Applications (GCMA)

Abstract

We consider the equation $u'+au=f$ on $\mathbb{R}$, where $a>0$ and $f$ is a real-valued Bohr almost periodic function, together with the backward Euler scheme

$$ \frac{U_n^h-U_{n-1}^h}{h}+aU_n^h=f(nh), \qquad n\in\mathbb{Z},\quad h>0. $$ Let $u_f$ and $U_{h,f}$ denote the respective bounded solutions. Both are almost periodic. We determine the least constant $C_a(h)$ such that $$ \sup_{n\in\mathbb{Z}}|U_{h,f}(n)-u_f(nh)| \leq C_a(h)\,\operatorname{Lip}(f) $$ for every Lipschitz almost periodic forcing. This constant is the integral of the absolute difference between the tails of the continuous exponential kernel and the discrete geometric kernel. It admits a finite-sum formula and satisfies $$ \lim_{h\downarrow0}\frac{C_a(h)}{h} =\frac{1-e^{-2}}{2a}. $$ The bound is sharp already within the class of periodic Lipschitz forcings. The proof uses a tail integration identity and a periodic approximation of its extremizers. For each fixed almost periodic forcing, the discrete bounded solution converges uniformly on the sampling lattice to the continuous one. In contrast, the supremum of the error over the unit ball of $\operatorname{AP}(\mathbb{R};\mathbb{R})$ equals $2/a$ for every $h>0$. A two-frequency example illustrates the sharp Lipschitz estimate and its relation to the individual Fourier modes.