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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|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====
<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>