# Questions tagged [poisson-summation-formula]

For questions dealing with the Poisson Summation Formula

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### Formal Poisson summation formula

I want to prove the following equality: $$\lim_{a\to -\infty,b\to \infty}\sum_{n=a}^b \frac {\sin \pi (c+n)}{\pi (c+n)}=1,\text{ for any }c\in \mathbb R.$$ All solutions I found directs me to an ...
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### Can I change the order of integral and summation in multidimensional Poisson summation?

Let $w$ be a smooth real function of compact support. Let $f$ be a real continuous function. Then the $n$ dimensional Poisson summation formula gives us \begin{eqnarray} &&\sum_{\substack{ \...
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### Confusion about applying Poisson's Sum Formula

I have quite a bit of confusion about Poisson's Sum Formula (PSF). With the standard definition of the Fourier Transform (FT), \begin{align} \hat{f}(\xi) &= \int_{-\infty}^\infty f(x)e^{-2\pi i x ...
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### Evaluation of a sum by means of Poisson sum formula and digamma function

I have the following series: $$\sum_{n=-\infty}^{\infty}\frac{1}{(2n+1)^2\pi^2+a^2}=\frac{1}{2a}\tanh\left(\frac{a}{2}\right)$$ and on the text it is written that it can be proven by means of either ...
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### Proving the relation: $\sum_{n\in\Bbb Z} \frac{2a}{a^2+4 \pi^2 n^2} = \sum_{n\in\Bbb Z} e^{-a \left\lvert n \right\rvert}$

I came across this problem in an exercise for Fourier analysis. I tried solving just $e^{−a|n|}$ to get the Fourier transform of a similar form as seen on the LHS because it looked familiar. But in ...
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### Proving the Partial Fraction Decomposition of the Hyperbolic Cotangent Function by using Poisson Summation

While skimming through the wonderful post What are some examples of colorful language in serious mathematics papers? on MathOverflow an example given by Ben Green aroused my curiosity. He referred to ...
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### G(k,X) is a modular form of weight k and character X

I'm trying to proof the transformation property of the Eisenstein series G(k,X) defined on page 17: https://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/978-3-540-74119-0_1/fulltext.pdf I already ...
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### Minkowski's Theorem by Harmonic Analysis

I am trying to properly write a proof of Minkowski's theorem in a self-contained way and understandable by (good) undergraduates. Theorem (Minkowski) Let $L$ be a lattice of $\mathbb{R}^n$ and ...
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### A Ramanujan sum involving $\sinh$

Today, in a personal communication, I was asked to prove the classical result $$\boxed{ \sum_{n\geq 1}\frac{(-1)^{n+1}}{n^3\sinh(\pi n)} = \frac{\pi^3}{360}}\tag{CR}$$ which I believe is due to ...
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### Find angle of a point a distance from another point on an ellipse defined by axial radius-es.

On an ellipse(p) defined by axial radius-es(rx, ry) where P1 & P2 are on p: Given rx, ry, α1 & Δ; Find α2 to P2. Illustration, because I can't post images yet, sorry for the link.
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### asymptotic expansion of a series related to $\cosh(x)$

Let the function $$F(x)=\frac{\pi\sinh(x)}{x}\sum_{n=-\infty}^{\infty}\frac{1}{\cosh\left[\frac{(2n+1)\pi^2}{2x}\right]}$$ where $\cosh(x)=\lambda\geq1$. For $\lambda\to1$, i.e., $x\to0$, what's ...
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Let's consider the function$F(x,y)$ defined by: $$F(x,y)= \sum_{n=1}^{\infty} f(nx) e^{-2 i \pi ny}$$ with $f(x)$ decreasing exponentially at infinity, $f(x)=0$ and $\int_{-\infty}^{\infty} f(|t|)dt=... 2answers 709 views ### Are the Euler-Maclaurin formula and the Poisson summation formula related? The Euler-Maclaurin formula, beautifully explained here by Justin Rheinstadter is expressed as: $$\sum_{i=m}^{n}f(i)=\int_{m}^nf(x)dx\;-\frac{1}{2}\left(f(n) - f(m)\right)\;+\sum_{k=1}^{p}\frac{B_{2k}... 0answers 88 views ### Clarification about Poisson summation formula I'm almost sure that I haven't understood correctly the Poisson formula$$\sum_{n=-\infty}^{+\infty}f(t-nT)=\frac{1}{T}\sum_{m=-\infty}^{+\infty}F\left(\frac{2\pi m}{T}\right) e^{\frac {-i2\pi m t}{T}... 1answer 422 views ### How can I prove that$X+Y$is a Poisson process with parameter$\lambda_X+\lambda_Y$? [duplicate] For 2 independent Poisson processes$X,Y$, with parameters$\lambda_X, \lambda_Y$respectively, how can I prove that$X+Y$is a Poisson process with parameter$\lambda_X+\lambda_Y$? To do this, I ... 2answers 62 views ### Find the summation formula of? I am trying to represent the following in summation form, Ex 1: let say the upper bound is 16, the lower bound is 1 The summation should be able to give the sum of 16 + 8 + 4 + 2 + 1 let say the ... 0answers 282 views ### Poisson summation formula for positive integers I am trying to evaluate the following expression for$\lambda \in \mathbb{R}$: $$f(\lambda)=\sum_{n=1}^{+\infty}e^{-i\lambda n}$$ My idea is to introduce an epsilon prescription, so I choose$\...
In our lecture we proved following version of the PSF: Assume $f \in L^1(\mathbb{R})$ and  \exists C,\varepsilon>0: \quad |f(t)|+|\hat{f}(t)|\le C(1+|t|)^{-1-\varepsilon} \quad \forall t\in\...