Defining periodic functions?

Question

Consider $f: (a,a+p] \rightarrow \mathbb{R}$.

What is the "formula" for the p-periodic function $g$ which has the property that $g(x + np) = f(x)$ for all $x$ in $(a, a+p]$? I am well aware of how it looks like graphically, but in what formal notation is this function written, as a piecewise function?

Answer 1

Given $x \in \mathbb{R}$, there exists a unique $y \in (a,a+p]$ such that $x = y + np$ for some $n$. We usually write $y = x - a \pmod p$. It is as though $y$ were the remainder of the "division of $x-a$ by $p$", with the exception that $p$ need not be an integer.

In this notation, the function $g$ becomes simply $g(x) = f(x-a \pmod p)$.

Answer 2

Let $\lceil x \rceil$ be the ceiling of $x$. Then $\lceil x/p \rceil < x/p \leq \lceil x/p \rceil +1$, thus $a < x+a-p\lceil x/p \rceil \leq a+p$. We can now define the $p$-periodic function $g$:

$$ g:\mathbb{R}\to\mathbb{R}, \;x\mapsto f(x+a-p\lceil x/p\rceil). $$