PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (200)
Orphanage
Unclass'd
Unproven (490)
Corrections (6)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: No information on entry rating
periodic extension (Definition)

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

  1. $ g$ is defined on $ \mathbb{R}$ except perhaps at points $ a+n(b-a)$, where $ n\in\mathbb{Z}$;
  2. $ g(x)=f(x)$ for all $ x\in (a,b)$, and
  3. $ g(x+n(a-b))=g(x)$ for all $ x\in (a,b)$ and all integers $ n$.

The best way to understand periodic extensions of a function is to look the graph of a periodic extension of a real-valued function. For example, let $ f(x)=x$ be defined on $ [-1,1]$. The graph of $ f$ looks like


\begin{pspicture} % latex2html id marker 79 (-6,-2)(6,2) \psaxes[Dx=9,Dy=2]{->}(... ....2,1.6){$y$} \psline(-1,-1)(1,1) \psdots[dotscale=1](1,1)(-1,-1) \end{pspicture}

Then the graph a periodic extension $ g$ of $ f$ may look like


\begin{pspicture} % latex2html id marker 86 (-6,-2)(6,2) \psaxes[Dx=9,Dy=2]{->}(... ...3,-1)(5,-1) \psdots[dotscale=1](1,1)(-1,1)(3,1)(-3,1)(5,1)(-5,1) \end{pspicture}

or look like


\begin{pspicture} % latex2html id marker 93 (-5.5,-2)(5.5,2) \psaxes[Dx=9,Dy=2]{... ...-5,1)(5,-1) \psdots[dotscale=1](-1,0)(1,0)(-3,0)(3,0)(-5,0)(5,0) \end{pspicture}

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(b-a)$ getting mapped to two distinct values $ f(a)$ and $ f(b)$, rendering $ g$ not well-defined. 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(b-a)$, where $ T=\lbrace x+n(a-b)\mid x\in S\rbrace$ and $ n\in\mathbb{Z}$;
  2. $ g(x)=f(x)$ for all $ x\in S-\lbrace a,b\rbrace$, and
  3. $ g(x+n(a-b))=g(x)$ for all $ x\in S-\lbrace a,b\rbrace$ 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
    1. either $ f(a)=f(b)$ when $ f$ is defined on a closed interval, or
    2. $ f(a+)=f(b-)$ when $ f$ is defined on an open interval, where $ f(a+)$ is the one-sided limit approaching $ a$ from the right, and $ f(b-)$ is the one-sided limit approaching $ b$ from the left.
    Simply define the periodic extension $ g$ so that either $ g(a)=f(a)$, or $ g(a)=f(a+)$. For example, the following graph

    \begin{pspicture} % latex2html id marker 144 (-6,-0.5)(6,2) \psaxes[Dx=9,Dy=2]{-... ...)(3,1) \psline(3,1)(4,0) \psline(4,0)(5,1) \psline(5,1)(5.5,0.5) \end{pspicture}
    is the graph of the continuous periodic extension of a function $ f_1$ given by $ f_1(x)=\vert x\vert$ defined on $ (-1,1)$, or a function $ f_2$ defined on $ (0,2)$, given by $ f_2(x)=x$ for $ 0<x\le 1$ and $ f_2(x)=2-x$ for $ 1\le 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$.

Bibliography

1
G.P. Tolstov, Fourier Series, Prentice-Hall, 1962.



"periodic extension" is owned by CWoo. [ full author list (2) ]
(view preamble)

View style:

See Also: periodic functions, triangular wave function
Log in to rate this entry.
(view current ratings)

Cross-references: axis, onto, projection, parallelogram, parallelepiped, easy to see, right, one-sided limit, iff, angles, trigonometric functions, rational numbers, proper subset, open interval, subset, closed interval, entire, domain, continuous, endpoints, well-defined, end points, odd integer, graph, integers, points, mean, line, interval, real, function
There is 1 reference to this entry.

This is version 14 of periodic extension, born on 2007-09-21, modified 2007-09-26.
Object id is 9955, canonical name is PeriodicExtension.
Accessed 467 times total.

Classification:
AMS MSC: 42A99 (Fourier analysis :: Fourier analysis in one variable :: Miscellaneous)
Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)