Sum of Replicative Functions is Replicative

Theorem

Let $f: \R \to \R$ and $g: \R \to \R$ be real functions.

Let $f$ and $g$ both be replicative functions.

Then the pointwise sum of $f$ and $g$ is also a replicative function.

$\ds \sum_{k \mathop = 0}^{n - 1} \map {\paren {f + g} } {x + \frac k n}$

$=$

$\ds \sum_{k \mathop = 0}^{n - 1} \paren {\map f {x + \frac k n} + \map g {x + \frac k n} }$

Definition of Pointwise Sum

$\ds $

$=$

$\ds \sum_{k \mathop = 0}^{n - 1} \map f {x + \frac k n} + \sum_{k \mathop = 0}^{n - 1} \map g {x + \frac k n}$

$\ds $

$=$

$\ds \map f {n x} + \map g {n x}$

$\ds $

$=$

$\ds \map {\paren {f + g} } {n x}$

Definition of Pointwise Sum

Hence the result by definition of replicative function.

$\blacksquare$