Questions tagged [poisson-summation-formula]

For questions dealing with the Poisson Summation Formula

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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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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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51 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 ...
2
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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=...
1
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1answer
28 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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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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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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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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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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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 ...
0
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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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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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40 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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36 views

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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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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0answers
26 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.
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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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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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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}{...