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)