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periodic extension
Let $f$ be a function defined on some real interval $[a,b]$. By a periodic extension of $f$ to the real line we mean a function $g$ such that
The best way to understand periodic extensions of a function is to look the graph of a periodic extension of a realvalued function. For example, let $f(x)=x$ be defined on $[1,1]$. The graph of $f$ looks like
\pspicture(6,2)(6,2) \psaxes[Dx=9,Dy=2]¿(0,0)(5.5,1.5)(5.5,1.5) \rput(5.8,0)$x$ \rput(0.2,1.6)$y$ \psline(1,1)(1,1) \psdots[dotscale=1](1,1)(1,1)
Then the graph a periodic extension $g$ of $f$ may look like
\pspicture(6,2)(6,2) \psaxes[Dx=9,Dy=2]¿(0,0)(5.5,1.5)(5.5,1.5) \rput(5.8,0)$x$ \rput(0.2,1.6)$y$ \psline(5.5,0.5)(5,1) \psline(5,1)(3,1) \psline(3,1)(1,1) \psline(1,1)(1,1) \psline(1,1)(3,1) \psline(3,1)(5,1) \psline(5,1)(5.5,0.5) \psdots[dotstyle=o,dotscale=1](1,1)(3,1)(1,1)(5,1)(3,1)(5,1) \psdots[dotscale=1](1,1)(1,1)(3,1)(3,1)(5,1)(5,1)
or look like
(5.5,2)(5.5,2) \psaxes[Dx=9,Dy=2]¿(0,0)(5.5,1.5)(5.5,1.5) \rput(5.8,0)$x$ \rput(0.2,1.6)$y$ \psline(5.5,0.5)(5,1) \psline(5,1)(3,1) \psline(3,1)(1,1) \psline(1,1)(1,1) \psline(1,1)(3,1) \psline(3,1)(5,1) \psline(5,1)(5.5,0.5) \psdots[dotstyle=o,dotscale=1](1,1)(1,1)(3,1)(1,1)(1,1)(3,1)(5,1)(3,1)(3,1)(5,1)(5,1)(5,1) \psdots[dotscale=1](1,0)(1,0)(3,0)(3,0)(5,0)(5,0)
Notice the two periodic extensions of $f$ are identical except at odd integer points on the $x$axis. The reason why we do not require $g$ to agree with $f$ on the end points of $[a,b]$ is because we do not know if $f(a)=f(b)$. If they do not agree, requiring that $g=f$ on all of $[a,b]$ may result in points $a+n(ba)$ getting mapped to two distinct values $f(a)$ and $f(b)$, rendering $g$ not welldefined. In fact, if $f$ does not agree on its endpoints, no periodic extensions of $f$ are continuous.
Notice, also, that the domain of function $f$ does not have to be the entire closed interval $[a,b]$. The domain of $f$ may very well be a subset $S\subseteq[a,b]$. For example, $f(x)=x$ may be a function defined on the open interval $(1,1)$. The two graphs above are again graphs of periodic extensions of $f$.
However, if $S$ is a proper subset of $[a,b]$ that is not the open interval $(a,b)$, then the definition of a periodic extension needs to be modified: $g$ is a periodic extension of $f$ defined on $S\subseteq[a,b]$ if
1. $g$ is defined on a subset $T\subseteq\mathbb{R}$ except perhaps at points $a+n(ba)$, where $T=\{x+n(ab)\mid x\in S\}$ and $n\in\mathbb{Z}$;
2. $g(x)=f(x)$ for all $x\in S\{a,b\}$, and
3. $g(x+n(ab))=g(x)$ for all $x\in S\{a,b\}$ and all integers $n$.
We generally assume that $a=\inf S$ and $b=\sup S$.
For example, if $f(x)=x$ for all rational numbers $x\in[1,1]$, then a periodic extension of $f$ has its domain the set of all rational numbers except perhaps at are odd integers.
Remarks.

Trigonometric functions defined on $\mathbb{R}$ are periodic extensions of the trigonometric functions defined for angles in the interval $[0,2\pi]$.

Suppose $f$ is defined either on a closed interval $[a,b]$ or an open interval $(a,b)$, $f$ has a continuous periodic extension (defined on all of $\mathbb{R}$) iff $f$ is continuous and that
(a) either $f(a)=f(b)$ when $f$ is defined on a closed interval, or
(b)
Simply define the periodic extension $g$ so that either $g(a)=f(a)$, or $g(a)=f(a+)$. For example, the following graph
\pspicture(6,0.5)(6,2) \psaxes[Dx=9,Dy=2]¿(0,0)(5.5,0.5)(5.5,1.5) \rput(5.8,0)$x$ \rput(0.2,1.6)$y$ \psline(5.5,0.5)(5,1) \psline(5,1)(4,0) \psline(4,0)(3,1) \psline(3,1)(2,0) \psline(2,0)(1,1) \psline(1,1)(0,0) \psline(0,0)(1,1) \psline(1,1)(2,0) \psline(2,0)(3,1) \psline(3,1)(4,0) \psline(4,0)(5,1) \psline(5,1)(5.5,0.5)
is the graph of the continuous periodic extension of a function $f_{1}$ given by $f_{1}(x)=x$ defined on $(1,1)$, or a function $f_{2}$ defined on $(0,2)$, given by $f_{2}(x)=x$ for $0<x\leq 1$ and $f_{2}(x)=2x$ for $1\leq x<2$. With $f_{1}$, we see that $f_{1}(1+)=f_{1}(1)=1$, while with $f_{2}$, we have $f_{2}(0+)=f_{2}(2)=0$.
It is easy to see that if a continuous periodic extension of a function exists, then it is unique.

Higher dimensional periodic extensions may also be defined for functions defined on a parallelepiped ($n$dimensional analog of a parallelogram). A periodic extension $g$ of a function $f$ defined on a parallelepiped is a function such that its projection $p_{i}(g)$ onto axis $i$ in $\mathbb{R}^{n}$ is a periodic extension of the projection $p_{i}(f)$ of $f$ onto axis $i$.
References
 1 G.P. Tolstov, Fourier Series, PrenticeHall, 1962.
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