Poisson sum formula

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A Poisson sum formula (Poisson will be boiled and comes English: Poisson summation formula) is a formula to insist on the infinity sum and the infinity sum of a certain function line of thing that Fourier transform did a function line being equal in mathematics. I was discovered by Simeon D'ni Poisson (Siméon Denis Poisson).

Proof

I am shown by the following expression transformation.

${\displaystyle {\begin{aligned}\sum _{k=-\infty }^{\infty }{\hat {f}}(k)&=\sum _{k=-\infty }^{\infty }{\bigg (}\int _{-\infty }^{\infty }f(x)\,e^{-i2\pi kx}dx{\bigg) }=\int _{-\infty }^{\infty }f(x)\underbrace {{\Bigg (}\sum _{k=-\infty }^{\infty }e^{-i2\pi kx}{\Bigg)}} _{\sum _{n=-\infty }^{\infty }\delta (x-n)}dx\\&=\sum _{n=-\infty }^{\infty }{\bigg (}\int _{-\infty }^{\infty }f(x)\,\delta (x-n)dx{\bigg) }=\sum _{n=-\infty }^{\infty }f(n)\end{aligned}}}$ 

Applied

It is applied to the proof in conjunction with テータ function, the Lehman zeta function.

Allied item

• Fourier transform
• テータ function
• Lehman zeta function
• Simeon D'ni Poisson

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This article is taken from the Japanese Wikipedia Poisson sum formula

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