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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 $\int_0^1 f(x)$.

Dan1618
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    Notice $F_N(0)$ is the usual Riemann sum of $f$ with left endpoints over $[0,1]$. Then the periodicity of $f$ tells you that the same is true for $F_N(x)$. So the limit is the constant $\int_0^1 f$, for all $x$. – Jose27 Jan 09 '22 at 17:10

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