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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?

Finanic Michelo
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2 Answers2

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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 $g(x) = f(x-a \pmod p)$.

Pedro M.
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  • Hello. Thank you for the answer. If $f(x) = x$, for example, then evidently there are points of discontinuity for $g$ at $x = a + np$. How is this reflected in the definition of $g$? – Finanic Michelo May 29 '15 at 13:11
  • @FinanicMichelo If $f$ is continuous, then $g$ is continuous if and only if $\lim_{x \searrow a} f(x) = f(a + p)$. In your example, this does not hold true, as $\lim_{x \searrow a} f(x) = a$ but $f(a+p) = a+p$. Hence, $g$ is not continuous. – Pedro M. May 29 '15 at 13:14
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    @FinanicMichelo The formula $g(x) = f(x-a \pmod p)$, although apparently simple, holds the potential for discontinuities. Afterall, it is the composition of $f$ with the discontinuous function $x \mapsto x-a \pmod p$. – Pedro M. May 29 '15 at 13:15
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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 $g$:

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

zuggg
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