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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Signal with finite length in time and frequency

Is it possible for a signal to have finite length in both the time domain and the frequency domain? Or does the finite length of one necessarily imply that the other has infinite length? (By "finite ...
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175 views

Finding the eigenvalues of the sum of circulant and diagonal matrices - What am I doing wrong?

Saw this question about the eigenvalues of the sum of circulant and diagonal matrices on MO and, since I recall my prof mentioned circulant matrices and Robert Gray's book, I thought I'd give it a ...
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378 views

Rate of Fourier decay of indicator functions

The Fourier transform of the indicator function of an interval $$\widehat{\chi}_{[a,b]}(\xi)=\int^b_{a} e^{i \xi x}dx=\frac{e^{i\xi b}-e^{i\xi a}}{i\xi}$$ has decay $O(|\xi|^{-1})$ as ...
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243 views

Complex numbers and Fourier transform

Here I am stuck in solving Fourier transform and the funny part is that I am stuck in the basics, in the complex part. I hope someone can help me solve this part. $$ 3 + 3 ( \cos \frac{4\pi}3 + j ...
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116 views

Is it true that a fourier transform of $f$ never vanishes if the translates of $f$ is $L^1(\mathbb{R})$

Let $f\in L^1(\mathbb{R})$ and let $V_f$ be the closed linear subspace of $L^1(\mathbb{R})$ generated by the translates $f(\cdot - y)$ of $f$. If $V_f=L^1(\mathbb{R})$, I want to show that $\hat{f}$ ...
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279 views

$\hat f_\lambda(x) = \hat f(x/\lambda)$

This is a small part of a larger problem I am trying to solve. This is stated as a basic property of the fourier transform. First we define for $f \in L^1(\mathbb{R}^d)$ and $\lambda \neq 0$, $$ \hat ...
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673 views

Computing the Gaussian integral with Fourier methods?

There are many proofs that $$\int_{-\infty}^\infty e^{-x^2} \, \mathrm dx = \sqrt{\pi}.$$ For example, using a change to polar coordinates, differentiation under the integral sign, and the theory ...
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432 views

Integrability of the Hilbert transform of a Schwartz function

Given a Schwartz function $f\colon\mathbb{R}\to\mathbb{R}$, define its Hilbert transform by $$(Hf)(x)=\frac{1}{\pi}\left(\int_{|t|\leq 1}\frac{f(x-t)-f(x)}{t}\,dt + \int_{|t|\geq ...
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1answer
144 views

Function of $C_0(\mathbb{R})$

I need to prove that $$g(x) = \text{p.v.} \int\limits_{-1/2}^{1/2}\frac{e^{-itx}}{t\cdot \ln{|t|}}dt $$ is function of $C_{0}(\mathbb{R})$. So, I need to prove that $$ ...
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1answer
173 views

Fourier transform and distibution beloning to S'

I need prove, that a distribution $$\langle F_f,\phi \rangle= p.v. \int\limits_{-1/2}^{1/2}\frac{\phi(t)}{t\cdot \ln{|t|}}\mathrm{d}t$$ belongs to $S^\prime$ (adjoint to Schwartz space) and I need ...
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179 views

Discrete Fourier transform of a particular sequence of real numbers

This may be an elementary question, but I don't know much about the discrete Fourier transforms (DFT). Suppose I have a sequence $\{x_n\}_{n=0}^{N-1}$ of $n$ real numbers such that $x_0\geq |x_n|$ ...
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384 views

How to use Fubini's theorem when integrating over Euclidean Space

I am currently studying the Fourier transform in Euclidean Space $\mathbb{R}^n$, using Knapp's book "Basic Real Analysis". Upon proving the property that the Fourier transform turns Convolution in ...
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1answer
163 views

Convolution of function

I need find $x^2\cdot e^{{-x^2}/2} * e^{{-x^2}/2}$. I used statements, that $\widehat{xf}=i \widehat{f}'$ and $\widehat{f*g}=\sqrt{2\pi}\widehat{f}\cdot \widehat{g}$. So, $\widehat{x^2\cdot ...
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1answer
173 views

Example of a function that is not the Fourier transform of any function of L1 (R).

Let $$ g(x) = v.p.\int_{-1/2}^{1/2}{e^{-itx}\over t\ln{|t|}}dt. $$ Please help me prove, that this function is not the Fourier transform of any function of $L_1(\mathbb{R})$.
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Fourier transform of $ 1/|x|^{k}$ [duplicate]

is ist possible to find the Fourier transform (direct and inverse ) of $ f(x)= \frac{1}{|x|^{k}} $ for $ k=1,2,3,......$ this function has a severe singularity at $x=0$ so i think this will exists ...
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87 views

question on the expansion of the function

For a given real number $c>0$ define functions $\left(\psi_k^c(\cdot)\right)_{k\ge0}$, as an eigenfunctions of the Sturm-Liouville operators $L_c$ defined $$ ...
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581 views

Fourier Transform of a frequency linearly modulated signal

I'm working on an oscillating signal whose trend can be modelled as a frequency linearly varying function. An example may be as follows: $$ \Gamma(t)=\sin(2\pi\nu(t)t) $$ with $$ \nu(t)=\nu_0 + at $$ ...
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468 views

What is the relationship between generalized functions and things like the Riesz representation theorem?

I just watched this video of Prof. Osgood's lecture on Fourier Transforms, and it seems to me that there's some connection between his talk of distributions (generalized functions) and the usual ...
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355 views

Discontinuous Fourier transforms?

What's an example (or even better a large class of examples) of an $L^2$ function whose Fourier transform is discontinuous?
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259 views

Fourier integral/ Fourier transformation of an oscillatory function with FFT

$f(x) = \cos(x^2)$ and $g(k) = \sqrt\pi \cos((\pi k)^2 - \pi/4)$ are a Fourier pair. I want to reproduce $g(k)$ by Fourier integrating $f(x)$ using FFT, i.e. approximating ...
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Truncation in singular integrals

After some thinking, I have a terrible headache caused by the following problem. Imagine we have a function $u \colon \mathbb{R}^n \to \mathbb{R}$ such that $u \in L^2(\mathbb{R}^n)$ ...
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269 views

Discrete Fourier transform real signals

The discrete Fourier transform is defined as: $$S(k)=\sum_{n=0}^{N-1}s(n)e^{-j\frac{2\pi}{N}kn}\quad k=0,...,N-1$$ I read that real signals $s(n)$ are: $$S(l)=S(N-l)^*$$ where $S(N-l)^*$ is the ...
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267 views

Heat equation asymptotic behavior

This was a question on one of my analysis finals, but I was unable to answer it. It seemed very interesting though. Suppose that $f(x, t)$ is a solution of the heat equation $\displaystyle ...
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1k views

Three Dimensional Fourier Transform of Radial Function without Bessel and Neumann

I am trying to compute the Fourier transform of $\frac1{|\mathbf{x}|^2+1}$ where $\mathbf{x}\in\mathbb{R}^3$. Just writing out the integral: ...
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210 views

Given fourier series,finding functions

4.How to prove that there is a continuous periodic function $f$ (with period $2\pi$), such that $$\hat{f}(n) = \log(n)/(n^{3/2}).$$ $n\neq 0$ and $\hat{f}(0) = -1$. I know only the basics of ...
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362 views

Fourier transform on $\mathbb{R}^3$

I've been stuck on this one homework problem for nearly a day now, so I'd be really thankful for any pointers. The problem is to find $$\int\int\int_{\mathbb{R}^3} \frac{1}{\mathbf{x}^2+1}e^{-2\pi i ...
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1answer
203 views

Reference for studying the method of stationary phase

I would like to learn about the stationary phase method, as part of the theory of Fourier Integral Operators. From what I can see so far, Hormander's book "The Analysis of Linear Partial Differential ...
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123 views

Fast Fourier Transform $\mathbf{\tilde{A}}=\mathbf{A}e^{i \theta(\mathbf{k})}$

I have a vector which takes form $\mathbf{\tilde{A}}=\mathbf{A}e^{i \theta(\mathbf{k})}$, where $\mathbf{k}$ is the frequency vector ($k^2=k_x^2 +k_y^2+k_z^2$), $i$ is unitary complex number, while ...
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157 views

Inverse Fourier transform of $\varphi(t) = \exp\left(\int_0 ^ 1 \frac{e^{itx} - 1}{x} \right)$?

I'm looking for the distribution whose Fourier transform is given by $$\varphi(t) = \exp\left(\int_0 ^ 1 \frac{e^{itx} - 1}{x} \right),$$ where as usual $\varphi(t) = \int_{- \infty} ^ \infty e^{itx} ...
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1answer
61 views

delaying signal

What does delaying a signal mean? Graphically? Mathematically? Is it, advancing to the next numbers, or using the previous numbers? Suppose i have $x[n] = \{0,1,2,3,4,5\}$ and i use $x[n-m]$ (example ...
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1answer
525 views

Fourier transform of Sin(x + Sin(x))

Does anybody happen to know what the Fourier transform of $f(x) = \sin(x + \sin(x))$ is? More generally, what is the Fourier transform of $g(x) = \sin(\phi_1 x + \mu \sin(\phi_2 x))$? ($\phi_1$, ...
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1answer
83 views

Extend convolution identity to $L^2$

I am trying to prove the following statement. if $f\in L^2$ and $g\in L^1$, $\mathcal{F}$ denotes fourier transformation,and$*$ denotes convolution, then ...
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1answer
120 views

Application of Fourier transformation

The problem requests to use Fourier transformation, which I totally have no clue how. It states as following: $u\in C^2_0$, prove $$\int_{\mathbb{R}^2}u_{xx}u_{yy}-u_{xy}^2 \,dx = 0$$ Any ...
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291 views

How to solve a linear PDEs with trigonometric function as coefficients

Is a general method for solving a system of linear partial differential equation with trigonometric function as coefficients exist ? For example something like that: $q$ is the unknown function, $2 ...
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1answer
37 views

estimate on exponential

I am currently reading a treatment on Fourier Series and the Heat trace, where the following estimate is stated: Suppose $\lambda > 0$ and $t \in (0,\infty)$. We have \begin{equation} ...
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37 views

every continuous signal being modelled as a function

Can every coninuous signal be modelled as a function, which then can be converted into a series of sine and consine functions with unique frequencies? And let us say that we have some arbitrary ...
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1answer
374 views

Fourier Transform of $ f(x) = \sin(x) \cdot e^{-|x|}$

Im trying to find the Fourier Transform of $ f(x) = \sin(x) \cdot e^{-|x|}$ I applied the standard formula and got to this point: $$\tilde{f(t)}= \frac{1}{\sqrt{2\pi}}\cdot ...
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1answer
1k views

Fourier transform of $\log x$ $ |x|^{s} $ and $\log|x| $

Can anyone provide or give an expression in the sense of distribution theory for the functions $|x|^{s} , \log|x| $? I mean I would like to evaluate the Fourier transform $ ...
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187 views

Is sampled absolutely integrable function absolutely summable?

Suppose I have function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that it's absolutely integrable: $\int_{\mathbb{R}}|f(x)|dx<\infty$. I am sampling function $f(x)$ with some period $T_s$. I am ...
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101 views

Fourier transform help

Find the Fourier tranform of $f(x)=x^2e^{-x^2}$ In a previous question when I found the Fourier transform of $f(x) =e^{-x^2}$, I used the formulas $F(f')=i\omega F$ and $F(xf)=iF'$. Will they be ...
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224 views

What function do I pick for $\sum_{n=1}^{\infty}(1/n^6)$? [duplicate]

Possible Duplicate: Computing $\zeta(6)=\sum\limits_{k=1}^\infty \frac1{k^6}$ with Fourier series. What function do I pick for the summation from $$\sum_{n =1}^{\infty}\frac{1}{n^6} \ ?$$ ...
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143 views

Product of Sidon sets

Let $G$ be a compact abelian group with dual $\Gamma$. Let $\Lambda \subset \Gamma$ a Sidon set (see the book of Rudin: Fourier Analysis on Groups for the definition). Consider the set ...
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420 views

Applying Plancherel's theorem to a simple function

Let $f(t) = \frac{1}{25}e^{-(t-11)^2}-\frac{1}{36}e^{-(t-13)^2}$. Using the Wiki definition of the Fourier transform pair, I calculated $\hat{f}$ in Mathematica as $$\hat{f}(\omega) = ...
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160 views

A Paley-Wiener like theorem in real-analysis

I try to identify conditions for the Fourier-transformation $\mathcal{F}(f)$ of some function $f \in L^1(\mathbb{R}^n)$ to be real-analytic. Namely I want to show that one of the following two ...
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Why the Fourier and Laplace transforms of the Heaviside (unit) step function do not match?

The Fourier transform of the Heaviside step function $u(t)$ is $\dfrac{1}{iω} + π δ(ω)$. The Laplace transform of the same function is $\dfrac{1}{s}$. I remember the proof came from derivatives and ...
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255 views

Properties of the Toeplitz matrices formed by square-summable sequence (as opposed to absolutely summable)

I've been reading a wonderful monograph by Robert Gray on the Toeplitz and circulant matrices and am curious about the assumption (4.3) of absolute summability of the sequences $\{t_k\}$ that form the ...
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400 views

Long integer multiplication using FFT in integer rings

I would like to perform long integer (~= polynomial) multiplication using the FFT or its direct analogue, but never leave integer rings. Please excuse in advance all my mistakes in formulation and ...
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2k views

white noise has flat power spectral density.

I am trying to prove that the white noise has constant power spectral density using matlab but the amplitude of the spectrum looks like random amplitude. can anyone tell me why? here is my code. ...
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91 views

A basic estimate for Sobolev spaces

Here is a statement that I came upon whilst studying Sobolev spaces, which I cannot quite fill in the gaps: If $s>t>u$ then we can estimate: \begin{equation} (1 + |\xi|)^{2t} \leq \varepsilon ...
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1answer
64 views

Primitive $2^k$-th roots of unity in $GF(p)$

While debugging an NTT implementation I've noticed something. Looks like if I have a primitive $(n = 2^k)$-th root of unity $\omega$ in a $GF(p)$, then $\omega ^0 = -\omega ^{n/2} + p,$ $\omega ^1 = ...