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Questions tagged [poisson-summation-formula]

For questions dealing with the Poisson Summation Formula

1
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1answer
46 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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0answers
15 views

a formulae of summation of exponential series till a finite term?

Is there a formulae in which I plug in the value of x and get the summation of exponential series till a finite n, or do I have to compute it the usual way?
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0answers
99 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 ...
2
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1answer
67 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 ...
1
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0answers
37 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 ...
4
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0answers
62 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 ...
3
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1answer
53 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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0answers
55 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}...
2
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2answers
599 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 \...
20
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3answers
413 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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0answers
25 views

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.
0
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1answer
121 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 ...
2
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0answers
85 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=...
9
votes
2answers
472 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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0answers
74 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}...
0
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1answer
289 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 ...
0
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2answers
55 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 ...
5
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0answers
259 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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0answers
72 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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0answers
87 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}{...
2
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1answer
107 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_{...