Questions tagged [poisson-summation-formula]

For questions dealing with the Poisson Summation Formula

The Poisson Summation Formula expresses a relation between the Fourier Series Coefficients of a periodic function and the values of its Fourier Transform given by

$$\sum_{n=-\infty}^\infty f(x+n)~=~\sum_{k=-\infty}^\infty e^{2\pi i kx}\int_{-\infty}^\infty f(t)e^{-2\pi ikt}\mathrm dt$$

53 questions
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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 Ramanujan. My proof can be found here and it is based…
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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) -…
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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 $\epsilon>0$, I then define a new function…
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What is the distributional limit $\displaystyle\lim_{N\to\infty} \sum_{n=0}^N e^{inx}$?

What is the distributional limit of $$\lim_{N\to\infty} \sum_{n=0}^N e^{inx} \; ?$$ If the summation is over $-N$ to $N$, the answer is $\sum_{m\in\mathbb Z}2\pi \delta(x-2\pi m)$, but what about the above one? Naive guess: If the sum is replaced by…
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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 $C$ a convex body, symmetric relatively to the origin…
TheStudent
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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 \left( \sum_{m ∈ ℤ} \phi(x+m) \right) \mathrm{e}^{-2…
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Poisson summation formula for possibly discontinuous function in Apostol

In Apostol, Mathematical Analysis, 2nd edition, Theorem 11.24 at page 332 states that Let $f$ be a nonnegative function such that the integral $\int_{-\infty}^{+\infty} f(x) dx$ exists as an improper Riemann integral. Assume also that $f$ increases…
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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 Poisson's summation formula to prove the following…
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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
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Infinite series Sum of zeroth order Bessel Functions of first kind

I am trying to find the upper bound of $$ \sum_{n \geq 1} J_0(an) J_0(bn) \sin(cn) \sin(dn) $$ where $J_0(x)$ is the zeroth order Bessel function of first kind, and $a,b \geq 0, \textit{ and } c,d \in \mathbb{R}$. I have tried to use Poisson…
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Summation formulas for integral transforms other than the Fourier

It seems to me that the Fourier transform harbours multiple useful results that allow one to sum an infinite series. The Poisson Summation Formula and Parseval's theorem. Also, Plancherel's theorem allows one to find the value of certain definite…
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Prove $\sum_{n=1}^{+\infty} \dfrac{1}{n^2} = \dfrac{\pi^2}{6}$ from $\lim_{a \to 0+} \dfrac{\pi}{a} \dfrac{1 + e^{-2 \pi a}}{1 - e^{-2 \pi a}}$?

Yes, I need to prove the solution to the Basel problem from a limit. The context is that we can combine the Fourier transform of $1/(n^2 + a^2)$ and the Poisson summation formula to show that (if $a > 0$) \begin{align} \sum_{n = -\infty}^{\infty}…
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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 sum $$\sum_{\boldsymbol{n}=\mathbf{1}}^{\infty}…
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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^{-2\pi i…
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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}^{\infty}f(2\pi…
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