Poisson summation formula

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)