Replicative Function of x minus Floor of x is Replicative

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Theorem

Let $f: \R \to \R$ be a real function.

Let $f$ be a replicative function.


Let $g: \R \to \R$ be the real function defined as:

$g \left({x}\right) = f \left({x - \left \lfloor{x}\right \rfloor}\right)$


Then $g$ is also a replicative function.


Proof

\(\displaystyle \sum_{k \mathop = 0}^{n - 1} g \left({x + \frac k n}\right)\) \(=\) \(\displaystyle \sum_{k \mathop = 0}^{n - 1} \left({f \left({x + \frac k n - \left \lfloor{x + \frac k n}\right \rfloor}\right)}\right)\)



Sources