Assume $f : \{1,\ldots,n\} \to \mathbb{C}$ satisfies $ |D^\ell f(i)| \leq C $ for all $i \in \{1,\ldots,n-\ell\}$ and all $\ell \in \{0, \ldots, k\}$ for some $k \in \mathbb{N}$. Here, $D$ denotes the finite difference operator $$ Df(i) := f(i+1) - f(i) \quad \text{for } i = 1,\ldots,n-1. $$
By analogy to the "continuous" approximation theory, one would expect we can approximate $f$ with much less than $n$ data. Are there any results on this topic?
Assume next that $f : \mathbb{Z} \to \mathbb{C}$ satisfies $|D^\ell f(i)| \leq C \, i^{-\ell-1}$. If $f$ was a function on $\mathbb{R}$, this property is called asymptotic smoothness and guarantees exponentially fast convergence for approximation by $hp$-finite elements. The question is again: are there any approximation results for such functions?
