We consider $d \in \mathbb{N}$. Suppose $q : \mathbb{Z}^d \to \mathbb{R}$ is a finite supported function and there's some $k \in \mathbb{N}$ such that for all $l \in \{ 1, ....k-1 \} $ and all $j_1 , ...j_l \in \{1, ...d\}$, we have $$ (*) \sum_{x = (x_1 ,x_2 , ...x_d ) \in \mathbb{Z}^d} x_{j_1} ....x_{j_2} q(x) = 0 $$
We define the kth order difference operator $\Gamma$ acting upon real functions $f$ by $$ \Gamma f (x) = \sum_{ y \in \mathbb{Z}^d} f(x+y) q(y), $$
The problem is
Prove that $\Gamma g_\varepsilon (0) = O( |\varepsilon|^k ) $ as $ \varepsilon \to 0$, where $g: \mathbb{R}^d \to \mathbb{R}$ is a smooth function, $\Gamma$ is a kth order difference operator, and $ g_\varepsilon (x) = g( \varepsilon x)$.
If we consider, for example, $d =1$, $k = 2$, and $ q (1) = q(-1) = 1/2$, $q(0) = 1$, and $q = 0$ otherwise. Then we obtain that the corresponding difference operator satisfies $$ \Gamma f (x) = \frac{ f(x+ 1)+ f(x - 1)}{2} - f(x) , $$ which is the second order difference of $f$ at $x$.In this particular case, we can use the Mean Value theorem to conclude the desired result. However, I have no idea how to generalize the result for any $q$ satisfying $(*)$. Could anyone provide a solution to it?