Questions tagged [poisson-summation-formula]

For questions dealing with the Poisson Summation Formula

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17 views

A right invariant measure on $H_1 \setminus H$ can be written as the product of right invariant measures on $ H_2 \setminus H$ and $H_1 \setminus H_2$

Suppose that $H$ is a locally compact, unimodular topological group. While studying the trace formula, I encountered the fact that I did not know which is that for any sequences $H_1 \subset H_2 \...
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1answer
42 views

Zero Inflated poisson distribution - sum of two random variables

Suppose X,Y are independent variables with $X \sim ZIP(\pi,\lambda)$ and $Y \sim ZIP(\pi,\lambda)$. The extended form of ZIP would be: $Pr(X=0) = \pi+(1-\pi)e^{-\lambda}$ $Pr(X=x) = (1-\pi)\frac{\...
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1answer
45 views

Proving Euler-Maclaurin approximation for even functions using Poisson summation

I'd like to ask for help for the following from my Advanced Mathematics for Physics class (6th semester): If $f(x)$ is a sufficiently regular and even function integrable in all $\mathbb{R}$, use ...
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30 views

Convergence of an adelic integral

Let $F$ be a Schwarz-Bruhat function on $\mathbb A = \mathbb A_{\mathbb Q}$, i.e. a finite sum of products of the form $\prod\limits_{p \leq \infty} f_p$, where $f_{\infty}$ is a Schwarz function on $\...
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1answer
65 views

Obtain the $N$-point IDFT of $X(k)=\frac{3}{5-4 \cos \left(2 \pi \frac{k}{N}\right)}$

I want to obtain the $N$-point IDFT of $X(k)=\frac{3}{5-4 \cos \left(2 \pi \frac{k}{N}\right)}$. My idea was to go with the discrete Poisson summation formula. $$ x_{s}(n)=\sum_{m=-\infty}^{\infty} x(...
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74 views

Evaluate in close form $\sum_{{n}=\mathbf{1}}^{\infty} \frac{\cos (\boldsymbol{\eta} \boldsymbol{n})}{\boldsymbol{n}^{2}+\boldsymbol{b}^{2}}$

Let Parameters $\boldsymbol{b} \in \mathbb{R} \backslash\{\boldsymbol{0}\}$ and $\boldsymbol{\eta} \in[\mathbf{0}, \boldsymbol{\pi}] .$ I want to evaluate in close form using hyperbolic functions the ...
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1answer
58 views

Application of Poisson summation formula [duplicate]

I am currently self reading " Spectral theory of Riemann zeta function" by Yoichi Motohashi. The example is in first chapter and of poincare series. I want to know how the Poisson sum ...
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2answers
344 views

How do you find $E(X^3)$ of a Poisson Distribution? [closed]

I know the proof to find the variance of a Poisson Distribution, and I tried to use that to find $E(X^3)$, but I can't get it to work. Any help would be great!!
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1answer
27 views

maximum values of the Poisson distribution

good day Can anybody help me? thanks let $X\sim \operatorname{Poisson}(\lambda)$ 1.- For what $\lambda$ values the value of $P (X = i), i≥0$ is maximum 2.- Prove that: $E (X^n) = \lambda E[(X+1)^{n-1}...
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62 views

Square of Jacobi theta function is sum of hyperbolic secant?

I'm presently reading Henri Cohen's Introduction to Modular Forms (https://arxiv.org/pdf/1809.10907.pdf) and I'm trying to do exercise 1.5, which partially entails showing that: $T_2(a)\equiv\sum_{n=-\...
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15 views

Drawing the spike-train of an alias sampled frequency graphically using the poisson summation formula

I have a signal $x(t) = sin(500 \pi t)$ which is sampled by a frequency $f_{s} = 300Hz$, given this, determine the output signal. Now, Let's the get the basics out first. Obviously this breaks the ...
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40 views

Justifying swapping limits in Poisson formula proof.

I've been looking at Henri Darmon's lectures notes for modular forms, and in lecture 3 he proves the Poisson summation formula, but one step of the proof is that $$\int_0^1\sum_{m\in\mathbb{Z}}f(x+m)e^...
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1answer
136 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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1answer
35 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
141 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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3answers
1k 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
42 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
41 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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82 views

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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29 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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78 views

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
60 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
77 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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112 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
318 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 made me curious. He referred to the ...
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51 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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118 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
84 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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57 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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4answers
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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660 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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1answer
359 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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103 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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919 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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107 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
564 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
68 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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318 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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92 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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128 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
117 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_{...