Fourier analysis, also known as spectral analysis, encompasses all sorts of Fourier expansions, including Fourier series, Fourier transform and the discrete Fourier transform (and relatives). The non-commutative analog is (representation-theory).

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Different methods to compute $\sum\limits_{n=1}^\infty \frac{1}{n^2}$

As I have heard people did not trust Euler when he first discovered the formula $$\zeta(2)=\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler and he gave other proofs. I ...
3
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0answers
18 views

Calculate difficult Fourier Transform

I have to calculate a quite difficult Fourier Transform for my class $$\int_{-\infty}^\infty dw\frac{(\varGamma-iw)w^3}{(\varGamma-iw)^2+1}\frac{J_{1}(|wr|)}{|wr|}e^{-iwt}$$ $J_1$ is the normal Bessel ...
1
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1answer
21 views

Smoothness of Fourier transform of a measure

Is the Fourier transform of a finite Borel measure on $\mathbb{R}$ necessarily a smooth function?( $\widehat{\mu}(x)=\int_\mathbb{R}e^{-i\pi xy} d\mu(y)$)
5
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1answer
75 views

Bounds on the line for entire functions of exponential type

Let $f$ be an entire function on the complex plane $\mathbb C$, assume that $$|f(z)|\le e^{|z|}.$$ Does the property $$|f(x)|\le e^{-|x|}, \qquad x\in\mathbb R,$$ imply $f\equiv 0$? More generally, ...
2
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0answers
26 views

Space of multipliers on $C_0(\mathbb{R})$?

What is the space of multipliers (that is translation invariant bounded operators) on $C_0(\mathbb{R})$ (space of continuous functions vanishing at $\pm \infty$)? I suppose the answer should be the ...
1
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1answer
73 views

Questions about Fourier Series

I have recently started looking at the topoic of Fourier series. Consider the space of square integrable functions $L_{2}[0,2\pi]$. Where we define the inner product as $(f,g):= \int_{0}^{2\pi}fg dx$ ...
0
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1answer
22 views

Quantum fourier transformation Unitary proof.

I've found a bunch of these proofs online but I am having trouble understanding how the norm of the column/row is 1.
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1answer
14 views

Equivalence of Schwartz Space Definition

I've come across two definitions for what it means for a function to be in $\mathfrak S$, the Schwartz space. A function $f \in \mathfrak S$ if $f \in C^\infty$ and for all $j, k \geq 0$ integers, ...
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0answers
7 views

Mean value of discrete periodic signals

It is clear that a continuous periodic signal always takes at some point $x_m$ the absolute mean value. Then we could define the absolute mean value of the signal as the value that it takes at $x_m$. ...
2
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1answer
39 views

Fast fourier transform and nyquist frequency

Trying to figure out how to use Matlab to calculate the nyquist frequency of a signal. Given a function, lets say $y = 5\sin (2t + \pi /3) + \sin (t + \pi /2)$ for $t > 0$. How do we use a fft in ...
4
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1answer
268 views

A condition on Fourier transforms that implies absolute continuity

Is there any condition on the Fourier transforms of 2 positive measures $\sigma , \mu$ on the complex unit circle $\mathbb{T}$ that implies absolute continuity ( $\sigma\ll\mu$)?
2
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1answer
37 views

Spectral convergence of coefficients of a Fourier series

I have seen claims that if a smooth function $f(x)$ is represented by its Fourier series, $f(x)=\sum_{n=-\infty}^\infty a_ne^{i(nt)}$, then as $|n|\rightarrow\infty$, then $|a_n|\rightarrow 0$ ...
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0answers
27 views

Averaging and approximation

I read a paper reference at http://arxiv.org/pdf/1101.1764.pdf that if we average a set $V=\{V(t_0,\nu_0), V({t_1,\nu_1),..., V(t_n,\nu_n)}\}$; with $V(t_i,\nu_i)=e^{i\sigma(t_i,\nu_i)}$ then we can ...
2
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0answers
41 views

List of ODE's that can be solved by Fourier transform

I am teaching introductory level Fourier analysis and I want to give my students some basic and some not so basic examples of how to solve ordinary differential equations with the method of Fourier ...
6
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3answers
126 views

For which $s\in\mathbb R$, is $H^s(\mathbb T)$ a Banach algebra?

According to Theorem 4.39 in Adams & Fournier Sobolev Spaces: If $mp>n$, $m\in\mathbb N$, then $W^{m,p}(\Omega)$ is a Banach algebra, provided that $\Omega\subset\mathbb R^n$ satisfies the ...
3
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3answers
3k views

Understanding Fourier transform example in Matlab

I'm studying about Fourier series and transform and I get confused with the following Matlab example of Fourier transformation: ...
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0answers
22 views

An inverse Fourier transform of Riemann $\Xi(z)$ function

Riemann $\Xi(z)$ function is related to Riemann $\zeta(s)$ function via ($s=1/2+i z$): $$\Xi(z)=\frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\tag{1}$$ The functional equation for $\zeta(s)$ is ...
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0answers
25 views

On symbol of an improper intergal

In a paper of Ingham with the title "A note on Fourier transforms" (1933) (see http://jlms.oxfordjournals.org/content/s1-9/1/29.extract), he wrote $\int^{\infty} \frac{\epsilon(y)}{y}dy$, and he ...
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1answer
33 views

About the Fourier transform of the surface measure of the unit sphere

Let $d\sigma$ denote the surface measure on $\mathbb{S}^{n-1}$. To compute its Fourier transform $$ \hat{d\sigma}(\xi)=\int e^{-i x\cdot \xi}\, d\sigma(x), $$ a standard technique (cfr. Folland's ...
4
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2answers
28 views

Applying the Fourier transform to solve an ODE.

We are learning about fourier transfrms in class and I was wondering about solving the following ODE using this method. So, I want to solve the equation $u''(x)+u(x)=0$. Now, it is clear that a ...
3
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2answers
177 views

Different Versions of Fourier Series? What about Uniqueness?

Let $f(x)$ be a function, then for its Fourier series $$ f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos(nx) + b_n \sin(nx)) $$ I found two different definitions (both yielding different ...
2
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1answer
330 views

Fourier transforms of the Heavyside function and the absolute value function

I'm trying to obtain these two Fourier transforms. First of all, I'm using the following definition of Fourier transform: $$\cal{F}(f)(y)=\int_{-\infty}^{\infty}f(x)e^{-ixy}\;dx$$ What I have so ...
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0answers
24 views

Find a compactly supported function which concerns the Fourier transform

I have a function $f:\mathbb{R}\to\mathbb{C}$ belonging to $L^2(\mathbb{R})$ such that $\left|f(x)\right|\le e^{-|x|^{\gamma}}$ for all $x\in\mathbb{R}$ ($0<\gamma<1$), and the support ...
0
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2answers
30 views

Solving a simple integral by derivating w.r.t. to constants

In the following notes on the solution of the Wave equation by Separation of Variables, in Example 2 the following derivation is given \begin{align*} \int_0^1 x \sin(k\pi x) d x & = \int_0^1 ...
4
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0answers
46 views

Is there a cyclic vector for $-\frac{d^{2}}{dx^{2}}$ on $L^{2}[0,2\pi]$ with periodic conditions?

Let $\mathcal{H}=L^{2}[0,2\pi]$, and let $L=-\frac{d^{2}}{dx^{2}}$ on the domain $\mathcal{D}(L)$ consisting of twice absolutely continuous functions $f$ on $[0,2\pi]$ with $f''\in\mathcal{H}$ and ...
0
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1answer
23 views

Exercise about Fourier transform from Rudin's book

Exercise 4 in chapter 9 of real and complex analysis: Give examples of $f\in L^2$ such that $f\notin L^1$ but $\hat{f}\in L^1$. Under what circumstances can this happen? I know function $\frac{\sin ...
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0answers
23 views

Fourier transform and conjugate variables

When you make a Fouriertransform of a function of time $f(t)$, it is said that it's Fouriertransform is a function of frequency $\widetilde{f}(\omega)$. The same argument goes for position and ...
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1answer
43 views

Mistake in evaluation of $\int_\mathbb{R} \frac{1}{1+t^4}e^{-itx}dt$

I am trying to use complex contour integration to calculate $\int_\mathbb{R} \frac{1}{1+t^4}e^{-itx}dt$. I have done the full calculation for $x>0$ and ended up with a function that is not ...
3
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0answers
39 views

Reference request for Fourier analysis on local fields

I am studing Class field theory. I need a good reference books, notes e.t.c which explains the following topics : Ideles and ideals, haar volume measure and integration on local fields, Fourier ...
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0answers
20 views

Questions on Heath-Brown's paper “Kummer’s Conjecture for Cubic Gauss Sums”

On page 21 in Heath-Brown's paper "Kummer’s Conjecture for Cubic Gauss Sums" (http://eprints.maths.ox.ac.uk/158/1/kummer.pdf), a formula says $$\sum_{j\in \mathbb{Z}[\omega]}f(j)=\sum_{k\in ...
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0answers
15 views

the partial derivative of poisson kernel w.r.t. theta tends to 0 as r tend to 1?

the partial derivative of poisson kernel w.r.t. theta tends to 0 as r tend to 1? how to show it ,thanks!!! seem it diverge,because we have a 'n' term after diff.?
1
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1answer
39 views

Fourier transform (properties)

I have a function $f$ such that $|f(x)|\leq e^{-x^2/2}$ hence in $\mathcal{L}^2(\mathbb{R})\cap\mathcal{L}^1(\mathbb{R})$ and thus we can compute the Fourier transform $$\hat{f} (\xi) = ...
1
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1answer
16 views

How to show a Fejér kernel is a good kernal??

I can prove the other two properties,but I cant show that the integration of the modulus of Fejér kernel is bdd,that is $\int$ |$K_n$|$\leq $ $M$ $for$ $all$ $n$ $\geq$$1$
2
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2answers
40 views

Fourier series, instantly determining $b_n$ once $a_n$ is found.

Find the Fourier series of the following function: $f(x) = \left\{\begin{align} 1+x,\quad -1\lt x \lt 0 \\ 1-x,\;\;\;\quad 0\lt x \lt 1\end{align} \right.$ $f(x+2) = f(x),\quad\quad -\infty \lt x ...
3
votes
1answer
82 views

Closure of Schwartz space in homogeneous Besov space

Let $\dot{B}^s_{\infty,\infty}(\mathbb{R}^d)$ denote the homogeneous Besov space of order $s$ with second and third index $\infty$, i. e. the homogeneous Zygmund space. Let $\mathcal{S}(\mathbb{R}^d)$ ...
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1answer
35 views

Setting up my Fourier series for $B_n$

Related but not necessary to know: here Looking at the temperature distribution in an infinitely long cylinder of metal with insulated sides and initial temperature distribution $f(x)= ...
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0answers
30 views

Proving that any continuous homomorphism of $\mathbb{R}/(2\pi\mathbb{Z})$ int0 $T$* is neccesarily an exponential function

This is an exercise form Katznelson's book on Harmonic Analysis, so I want to solve it using his hint. T* here denotes the multiplicative group of units of complex numbers of unit norm. That is to ...
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1answer
43 views

How did Fourier series lead to the development of rigorous analysis?

Once I've heard that the studies of Fourier series have lead to rigorous definitions of such concepts as function, convergence, integral, limit. And also that Cantor's study of Fourier series led him ...
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1answer
27 views

Fourier series representation of $\sin^4 x$

I tried solving for fourier coefficients of Fourier series for the multiples of fundamental frequency $\omega_0=2$. So $F_n=\int_0^{\pi} \sin^4 x \, e^{-i2nx} dx$. And my calculator says answer should ...
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0answers
33 views

Fourier transform with the function in denominator

I'm trying to do a Fourier transform of this function $$\frac{\bigtriangleup f(\textbf{x})}{1+f(\textbf{x})}$$ in terms of $\mathcal{F}(f(\textbf{x}))$. (Just like here ...
0
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1answer
26 views

$\int\limits_{\mathbb{R}} e^{-|x|}e^{-ix\xi}dx$

I can't compute this $\int\limits_{\mathbb{R}} e^{-|x|}e^{-ix\xi}dx$. I have separate it into 2 integrals but i can't continue.
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1answer
30 views

Find inverse Laplace transform of $H(s)=\frac8{s^4+4}$

How can we find the inverse Laplace transform of the function $$H(s)=\frac8{s^4+4}?$$
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0answers
19 views

How to determine if someone understands Fourier transforms

I barely understand Fourier transforms. I'm looking for a simple test or Q&A that when used would tell me if someone else understood Fourier transforms. How would I know if the person genuinely ...
20
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1answer
261 views

Using Fourier Series to compute sums

I have just started learning the basics of Fourier series and have some doubts about it. I am aware that Fourier series can be used to compute infinite sums. For example, $\zeta(2)$ and $\eta(2)$ can ...
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0answers
24 views

Polynomial division/deflation with FFT

There is a need to divide a polynomial $p(x)$ by polynomial $q(x)$, whereas it is known that the remainder will be zero (i.e. the question is about polynomial deflation). A known method is to use the ...
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0answers
24 views

White noise, how is its definition sensical

White noise is defined as as noise containing all frequencies. Now, consider the inverse fourier transform of white noise, $R$ being the fourier transoform of the noise: $$\int_{-\infty}^\infty R ...
1
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1answer
32 views

Fourier Transform solution of $\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} - u$

Let $u(x,t)$ solve the partial differential equation $$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} - u$$ where $x,t\in\mathbb{R}$ with $t>0$ and initial condition $u(x,0) = ...
0
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1answer
35 views

sinc in 2d: how to interprete this in spatial domain?

The following two images are the ideal low pass filter in the frequency domain. As you can see, the origin (low frequency component), can pass through this filter while the high frequency are blocked. ...
2
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1answer
34 views

Help solving $\frac{\partial^3u}{\partial x^3}=\frac{\partial u}{\partial t}$ using Fourier transforms

I am trying to solve $$\frac{\partial^3u}{\partial x^3}=\frac{\partial u}{\partial t}$$ $x\in\mathbb{R},\:t >0$. Subject to the conditions $u(x,0)=f(x),\:u,\:\frac{\partial u}{\partial ...
2
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0answers
59 views

Mellin transform on $\mathbb{Z}[\omega]$

Let $\omega=\frac{-1+i\sqrt{3}}{2}$ be a complex cube root of unity. The Eisenstein integers $\mathbb{Z}[\omega]$ (a unique factorization domain) are of the forms $a+b\omega$ where $a$ and $b$ are ...