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The formula
 
The formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073360/p0733601.png" /></td> </tr></table>
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$$
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\sum _ {k = - \infty } ^ { +\infty }
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g ( 2 k \pi )  = \
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\sum _ {k = - \infty } ^ { +\infty }
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\frac{1}{2 \pi }
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\int\limits _ {- \infty } ^ { +\infty }
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g ( x) e ^ {- i k x }  d x .
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$$
  
The Poisson summation formula holds if, for example, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073360/p0733602.png" /> is absolutely integrable on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073360/p0733603.png" />, has bounded variation and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073360/p0733604.png" />. The Poisson summation formula can also be written in the form
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The Poisson summation formula holds if, for example, the function $  g $
 +
is absolutely integrable on the interval $  ( - \infty , + \infty ) $,  
 +
has bounded variation and $  2 g ( x) = g ( x + 0 ) + g ( x - 0 ) $.  
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The Poisson summation formula can also be written in the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073360/p0733605.png" /></td> </tr></table>
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$$
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\sqrt {a } \sum _ {k = - \infty } ^ { +\infty } g ( a k )  = \
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\sqrt {b } \sum _ {k = - \infty } ^ { +\infty } \chi ( b k ) ,
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$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073360/p0733606.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073360/p0733607.png" /> are any two positive numbers satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073360/p0733608.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073360/p0733609.png" /> is the [[Fourier transform|Fourier transform]] of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073360/p07336010.png" />:
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where $  a $
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and $  b $
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are any two positive numbers satisfying the condition $  a b = 2 \pi $,  
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and $  \chi $
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is the [[Fourier transform|Fourier transform]] of the function $  g $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073360/p07336011.png" /></td> </tr></table>
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$$
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\chi ( u)  = \frac{1}{\sqrt {2 \pi } } \int\limits _ {- \infty } ^ {+ \infty } g(x) e ^ {- i u x }  d x .
 +
$$
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Zygmund,   "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.C. Titchmarsh,   "Introduction to the theory of Fourier integrals" , Oxford Univ. Press  (1948)</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  A. Zygmund, "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  E.C. Titchmarsh, "Introduction to the theory of Fourier integrals" , Oxford Univ. Press  (1948)</TD></TR>
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</table>

Latest revision as of 20:15, 16 January 2024


The formula

$$ \sum _ {k = - \infty } ^ { +\infty } g ( 2 k \pi ) = \ \sum _ {k = - \infty } ^ { +\infty } \frac{1}{2 \pi } \int\limits _ {- \infty } ^ { +\infty } g ( x) e ^ {- i k x } d x . $$

The Poisson summation formula holds if, for example, the function $ g $ is absolutely integrable on the interval $ ( - \infty , + \infty ) $, has bounded variation and $ 2 g ( x) = g ( x + 0 ) + g ( x - 0 ) $. The Poisson summation formula can also be written in the form

$$ \sqrt {a } \sum _ {k = - \infty } ^ { +\infty } g ( a k ) = \ \sqrt {b } \sum _ {k = - \infty } ^ { +\infty } \chi ( b k ) , $$

where $ a $ and $ b $ are any two positive numbers satisfying the condition $ a b = 2 \pi $, and $ \chi $ is the Fourier transform of the function $ g $:

$$ \chi ( u) = \frac{1}{\sqrt {2 \pi } } \int\limits _ {- \infty } ^ {+ \infty } g(x) e ^ {- i u x } d x . $$

References

[1] A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988)
[2] E.C. Titchmarsh, "Introduction to the theory of Fourier integrals" , Oxford Univ. Press (1948)
How to Cite This Entry:
Poisson summation formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poisson_summation_formula&oldid=13421
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article