Let $f:[0,1]\rightarrow \mathbb{R}$ be a function and consider the points $t=1,\dotsc,T$. Given an integer $\theta>0$, suppose I observe the values of the differences $$f_{\Delta_{k\theta}}(t)\equiv f\bigg(\frac{t+k\theta}{T}\bigg)-f\bigg(\frac{t}{T}\bigg),\forall k\in\mathbb{Z},$$
as long as $0\leq \frac{t+k\theta}{T}\leq 1$.
I am thinking how could I use these differences to estimate the values of $f$ on $\{t/T\}_{t=1}^T$. As $T$ grows, I noted that the points $\{t/T\}_{t=1}^T$ are getting "dense" on $[0,1]$. Maybe I should use this asymptotics to obtain some kind of approximation.
Toy example: Suppose $T=6$ and $\theta=2$. So I observe $f_{\Delta_2} (1)=f(3/6)-f(1/6),f_{\Delta_2}(3)=f(5/6)-f(3/6),f_{\Delta_4}(1)$, as well as $f_{\Delta_2} (2),f_{\Delta_2} (4)$ and $f_{\Delta_4} (2)$. How to approximate $f(1/6),\dotsc,f(6/6)$? Next, what if $T\rightarrow \infty$? the proposed method converges to $f$?
I know I didn't assume much about $f$. Maybe differentiability or something else are needed.