Questions tagged [poisson-summation-formula]

For questions dealing with the Poisson Summation Formula

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20 views

Discrete-time fourier transform as a periodic sum of fourier transform for different sampling

According to Wikipedia, for any function $x(t)$ with the fourier transform of $\hat x(t) = X(f)$, we can generate a discrete sample by "sampling" $x(t)$ at $T$-separated values to get a sequence $x[n] ...
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3answers
514 views

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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14 views

Simplifying Summation That Occurs In Poisson Counting Process

The question of how to compute the probability of an even or odd value occurring in the Poisson counting process has been answered on here in several places, but there's a step in the solution that I ...
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1answer
31 views

Textbook reccommendations

I'm looking to find an easy lay-man explanation of the Poisson Summation Formula and a few extra questions on the same. It would be great if someone could recommend a few to me! Thanks
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1answer
46 views

Evaluate $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2}$ with Poisson summation formula

This post (Closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$.) gives a closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$ with $b\gt0.$ And the result is $$\sum_{n\in\...
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2answers
112 views

Evaluating $\sum_{k=1}^{\infty}\left(\frac{\sin(tk)}{k}\right)^2$

Using Poisson Summation Formula, how do you evaluate the following infinite sum $\sum_{k=1}^{\infty}\left(\frac{\sin(tk)}{k}\right)^2$? The Poisson Summation Formula states that: $\sum_{k=-\infty}^{\...
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1answer
30 views

Transformation formula for Theta-series

I am currently reading Weil's book : "Elliptic Functions According to Eisenstein and Kronecker" and in page 56 he uses the well-known transformation formula for theta series $$\sum\limits_{\mu} ...
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1answer
29 views

Binomial distribution considering starting-a-set-advantage

I am making a model in Matlab that calculates the winning probability of darts player A against darts player B with help of a paper (see bottom line). I have been able to implement the calculation of ...
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0answers
5 views

Anti-Poisson or alternate poisson summation?

is there exist the alternating analogue to Poisson sum formula ?? i mean $$ \sum_{m=-\infty}^{\infty} (-1)^{m}f(m) = \sum_{m=-\infty}^{\infty}F(2\pi m) $$ here $ F(u)= \int_{-\infty}^{\infty} dx f(...
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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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24 views

Path Integral on a circle: extension of integral range to the real line.

In "Path integrals in physics, vol.1. Stochastic processes and quantum mechanics, Chaichian M., Demichev A", eq-(2.4.43) they have: $$\langle \phi,t|\phi_0,t_0\rangle_{\text{circle}}\approx\prod_{n=1}^...
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Question on Poisson summation formula and Fourier transform

I have a smooth function compact support $f(x,y)$. Then the Poisson summation formula gives $$ \sum_{n_1, n_2 \in \mathbb{Z}} f(n_1, n_2) = \sum_{m_1, m_2} \int_{\mathbb{R}^2} f(z_1,z_2) e^{- 2 \pi ...
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1answer
52 views

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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1answer
71 views

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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108 views

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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1answer
208 views

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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48 views

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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99 views

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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1answer
69 views

Probability density of the fractional part of |σZ| as σ→∞?

Consider the following probability density function parameterized by $\sigma>0$: $$g_\sigma(x)=2\sum_\limits{n=0}^\infty\phi_\sigma(x+n), \quad 0<x<1$$ where $$\phi_\sigma(t)={1\over \sigma\...
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56 views

Evaluating $\displaystyle\int_{-\infty+iy}^{\infty+iy}(cv)^{-k}e\left(\frac{-m}{c^2v}-nv\right)dv$.

So the book I'm reading tells me to derive \begin{align*} \mathcal{J}_c(m,n)&=\displaystyle\int_{-\infty+iy}^{\infty+iy}(cv)^{-k}e\left(\frac{-m}{c^2v}-nv\right)dv\\ &=\displaystyle\frac{2\pi}...
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2answers
2k views

Understanding Proof of Poisson Summation Formula

Consider a proof from a textbook on Harmonic Analysis: Note that $\mathcal{S}(ℝ)$ denotes the Schwartz Space. Question 1: Why does the top left formula in the proof start out as: $$ \int_0^1 \...
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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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1answer
237 views

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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98 views

Poisson summation formula and Fourier series

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=...
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2answers
720 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}...
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89 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}...
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1answer
433 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 ...
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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 ...
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285 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 $\...
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83 views

weaker assumptions on Poisson summation formula

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\...
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105 views

Elementary question about hypothesis for the Poisson Summation Formula

Updated: Clarified the question. I'm stuck in what must be a basic misunderstanding of the necessary hypotheses for the Poisson Summation Formula. Suppose $f(x) = \frac{1}{\sqrt{2\pi}}e^{\frac{-x^2}{...
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1answer
114 views

Rudin's Real & Complex - Q9.11 (Fourier)

I have solved most of Question 9.11 of Big Rudin : Find conditions on $f$ and/or $\widehat{f}$ which ensure the correctness of the following formal argument : If $\varphi(t) ~=~ \frac{1}{2\pi}\int_{...