I have a signal $s(t)$ defined over $t\in[0, T]$ that is assumed to be smooth up to order $d > 1$, i.e., its $d$th order derivative is continuous (and bounded by say $\epsilon$). Then I sample this signal with a sampling rate of $f_s$ (a sampling time $T_s = 1/f_s$) to obtain $s[n]=s(nT_s)$.
1- I'm looking for a way to express this problem and relate the order $d$ derivative bounds on $s(t)$ ($\epsilon$ or other bounds on its lower order derivatives, if needed) to the concept of local or global Lipschitz continuity. The relation seems trivial for $d=1$, by definition. How about $d>1$? Does there exist a thing like "higher-order Lipschitz continuity"?
2- How can I relate the first, second, or higher order sample difference operators such as $D_1(s[n]) = s[n] - s[n-1]$ or $D_2(s[n]) = s[n] - 2s[n-1] + s[n-2]$, etc. to the concept of Lipschitz continuity?
(sorry if the problem is not rigorously stated. I'm not a mathematician; I have a signal processing background)
Thanks!