Suppose I have a function $f$ such that $f(x) = f(x+1)$. We can assume its continuous (although I really just want $f$ to be in $L^1$ or $L^2$). Consider the sequence of functions
$$F_N(x) = \frac{1}{N}\sum_{n=0}^{N-1} f(x+\frac{n}{N})$$
Do we have $F_N \to f$ $L^1$ or $L^2$ or point-wise? Is there any connection between $f$ and $F_N$?
My question is inspired by the following post
Fejér's Theorem (Problem in Rudin)
which was able to relate something like $F_N$ to $\int_0^1 f(x)$.