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Difference between revisions of "Poisson summation formula"

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$$  
 
$$  
\sum _ {k = - \infty } ^ { {+ \infty }
+
\sum _ {k = - \infty } ^ { +\infty }
 
g ( 2 k \pi )  = \  
 
g ( 2 k \pi )  = \  
\sum _ {k = - \infty } ^ { {+ \infty }
+
\sum _ {k = - \infty } ^ { +\infty }
  
 
\frac{1}{2 \pi }
 
\frac{1}{2 \pi }
  
\int\limits _ {- \infty } ^ { {+ \infty }
+
\int\limits _ {- \infty } ^ { +\infty }
 
g ( x) e ^ {- i k x }  d x .
 
g ( x) e ^ {- i k x }  d x .
 
$$
 
$$
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$$  
 
$$  
\sqrt {a } \sum _ {k = - \infty } ^ { {+ \infty } g ( a k )  = \  
+
\sqrt {a } \sum _ {k = - \infty } ^ { +\infty } g ( a k )  = \  
\sqrt {b } \sum _ {k = - \infty } ^ { {+ \infty } \chi ( b k ) ,
+
\sqrt {b } \sum _ {k = - \infty } ^ { +\infty } \chi ( b k ) ,
 
$$
 
$$
  
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$$  
 
$$  
\chi ( u)  = \
+
\chi ( u)  = \frac{1}{\sqrt {2 \pi } } \int\limits _ {- \infty } ^ {+ \infty } g(x) e ^ {- i u x }  d x .
 
 
\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>
+
<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>

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=48222
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article