Tagged Questions

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).

learn more… | top users | synonyms

2
votes
0answers
23 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 ...
1
vote
0answers
8 views

Integrals of compactly supported functions of positive type

Consider a continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$, supported on $[-1,1]$, of positive type. Assume $f(0) = 1$; what is the "largest" area $\int f\,dx$ that can be achieved? To be ...
0
votes
0answers
11 views

Approximating Averaging : Signal processing

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 ...
0
votes
0answers
16 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 ...
0
votes
0answers
22 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 ...
0
votes
0answers
5 views

$R_S (=K \cap A_{K,S})$ is a Dedekind domain

Let $K$ be a global field and let $S$ be a finite, nonempty set of places of $K$ containing the infinite ones. Show that $R_S (=K \cap A_{K,S})$, the ring of $ S-$ integers of $K$, is a Dedekind ...
4
votes
2answers
24 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 ...
2
votes
1answer
26 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 $|a_n|\rightarrow 0$ ...
3
votes
2answers
176 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 ...
1
vote
0answers
19 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 ...
1
vote
0answers
28 views

$\Bbb{R}/n\Bbb{Z}$ is isomorphic to $A_\Bbb{Q}/(\Bbb{Q}+C_n)$.

Let $A_\Bbb{Q}$ be the adele group of $\Bbb{Q}$. Let $C_n=\{x \in A_\Bbb{Q}: x_\infty=0 \text{ and }x_p \in p^{\operatorname{ord}_p(n)}\Bbb{Z}_p \text{ for prime }p\}$. I want to show that ...
1
vote
1answer
28 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 ...
0
votes
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 ...
0
votes
1answer
20 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 ...
1
vote
1answer
38 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 ...
2
votes
0answers
27 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 ...
0
votes
0answers
13 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.?
2
votes
0answers
38 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
votes
0answers
15 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 ...
1
vote
1answer
34 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
vote
1answer
13 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$
-5
votes
0answers
22 views

Fourier integral representation. [closed]

What is the Fourier integral representation of $$ \begin{cases} f(x)=0, & x<0 \\ \frac{1}{2}, & x=0 \\e^{-x}, & x>0 \end{cases}.$$
2
votes
2answers
32 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 ...
0
votes
0answers
17 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 ...
1
vote
1answer
41 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 ...
1
vote
0answers
21 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 ...
1
vote
0answers
31 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 ...
1
vote
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 ...
0
votes
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.
1
vote
1answer
29 views

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

How can we find the inverse Laplace transform of the function $$H(s)=\frac8{s^4+4}?$$
0
votes
0answers
17 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 ...
0
votes
0answers
23 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
vote
1answer
30 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
votes
0answers
17 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 ...
2
votes
1answer
31 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 ...
0
votes
1answer
34 views

Fourier transform and residue theorem

I want to calculate the fourier transform of the funcion: $$f(x)=\frac{1}{a^2+x^2}$$ So I did: $$\hat f(w)=\frac 1 {\sqrt{2\pi}} \int_{-\infty}^{\infty} \frac {e^{-iwx}}{a^2+x^2}dx$$ And I tried to ...
0
votes
1answer
21 views

An integral converges- Analysis Fourier

I want to show that if $k\geq2$ and $f\in C_{c}(\mathbb{R})\cap C^k(\mathbb{R})$ then $\int\limits_{\mathbb{R}} |\xi|^{κ-2}|\hat{f}(\xi)|dm(\xi)<\infty$. I can't understand what can i use for ...
2
votes
0answers
54 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 ...
0
votes
0answers
6 views

Find the spectral density of white noise $\omega$~(0,2)

I am find the spectral density of a white noise given by $\omega$ ~ (0,2). Could you help me to find it? Thank all.
0
votes
1answer
13 views

Simplification of a large sum obtained from the 1-D wave equation

I have acquired the sum below through Fourier, and was wondering if there was anyway to simplify it, since it is large and ugly. $$\sum \limits_{n=1}^\infty \frac{-2K_1}{n\pi} ...
0
votes
1answer
28 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. ...
0
votes
1answer
30 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)= ...
1
vote
0answers
28 views

Solve this integral with transformations

I have the integral $$ I= \int_0^\infty\frac{w^2dw}{(1+w^2)^4}$$ and I think I can solve it using the fourier transform and Plancherels formula. I can write Plancherels as: $$\int_{-\infty}^\infty ...
0
votes
1answer
27 views

Prove that $\hat{f}(n)={\frac{2}{\pi(1-4n^2)}}$, given that $f(x)=|sin(\pi x)|$

Prove that $$\hat{f}(n)={\frac{2}{\pi(1-4n^2)}},\ given\ thatf(x)=|sin(\pi x)||,\int_{0}^{1}sin\pi(x)dx={\frac{2}\pi}\\where\ \hat{f}(x)=\int_{0}^{1}f(x)e(-nx)dx. \ Use\ the\ fact\ ...
0
votes
1answer
32 views

Writing a function in terms of the rect and delta functions.

Say I have a function that is equal to 1 at two unit area squares. One is centered at $(-3,0)$ and the other at $(3,0)$. I am trying to find a formula for this function using only the rect function ...
0
votes
0answers
19 views

A Direct Proof of Representation Theorem for Positive Harmonic Functions in the Half Plane?

Does anyone know a direct proof of this representation theorem for non-negative harmonic functions in the half-plane that doesn't appeal to a similar result in the unit disk? Also, does anyone who ...
0
votes
1answer
28 views

Strange thing about Weak Maximum\Minimum Principle?

I feel confused about this problem. I think it is obvious using Weak Maximum\Minimum principles. Since for harmonic functions. If $\Omega$ is bounded and $u\in C^2(\Omega) \cap C^0(\overline ...
0
votes
1answer
43 views

Integral for $\frac{x}{x^2+1}cosx$

When computing Fourier transformation I came across these integral: $$ \int_{\Bbb R}\frac{x \cos x}{1+x^2}\;dx\text{ or } \int_{\Bbb R}\frac{x \sin x}{1+x^2}\;dx $$ Can anyone give me some hints on ...
0
votes
0answers
21 views

Inverse Fourier Transform of $\cos(c\omega t)$ and $\sin(c\omega t)$

I'm just needing a bit of help to understand the derivation of the inverse fourier transform of $\cos(c\omega t)$ and $\sin(c\omega t)$, in deriving D'Alembert's solution to the wave equation. I get ...
1
vote
1answer
38 views

Is it always the case that lower frequencies contribute the most in a Fourier series?

Is it always the case that lower frequencies contribute the most in a Fourier series? Or to put it in other words, in the equation: $$f(t)=a_0+\sum^\infty_{m=1} a_m\cos \left(\frac{2\pi mt}{T}\right) ...