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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Fourier representation for $\tan(x)$
Q: Which Fourier representation is suitable for $f(x) = \tan(x)$: Fourier trigonometric series, Fourier half-range expansion, or Fourier integral and why?
Well I searched and found that:
$\tan(x)$ ...
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1answer
24 views
continuous Fourier transform
For some reals $a$ and $b$, we have $h(t)=p\left(\cfrac{t-a}{b}\right)$. Also, $p(s)$ is real and even function of $s$.
By the transformation rules, how can I determine the continuous Fourier ...
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Fourier transform of $f(r,r',\theta,\theta')$
How can I calculate the FT of:
$$\sum_{n=-\infty}^{\infty}\,e^{in(\theta' -\theta)}\,f_n(r,r')=\sum_{n=-\infty}^{\infty}\,e^{in(\theta' -\theta)}\,\frac{J_n(\alpha r)J_n(\alpha r')}{[(\alpha ...
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Inverse fourier transform of a function which is a fundamental solution
Let $f:\mathbb{R}^3 \to \mathbb{R}$ be $f(x)=(1+|x|^2)^{-1}$.
I need to calculate $\mathcal{F}^{-1}(f)$.
I've proven that $f\in L^2(\mathbb{R}^3)$ and I know that the fourier transform is an ...
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1answer
38 views
Multiplication in $\mathcal D'(R)$.
I was reading in my text book that multiplication of elements of $\mathcal D'(R)$ is not a continuous function although it is defined and can be seen from the convergence of $\sin(nx)$ to $0$ as $n$ ...
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Proof the Cooley-Tukey splitting and merging.
This image below shows the splitting and merging of the matrices.
In the first step of the FFT. you split a matrix with $N^2$ adds and multiplies to $2$ matrices, with $2$ times $(N/2)^2$ adds and ...
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Intuitive way to understand the triangle spectrum?
Image on the top is in the time domain, image on the bottom is in the frequency domain.
What do we see $-2T$ and $2T$ on image in the time domain and why do we see $-\frac{1}{2T}$ and $\frac{1}{2T}$ ...
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62 views
Does convergence in $C(\Bbb R)$ imply convergence in $\mathcal D'(\Bbb R)$
Sequence of continuous functions $(f_k)$ which converge to $f$ in $C(\Bbb R)$. Does it imply that $f_k$ will converge to $f$ in $\mathcal D'(\Bbb R)$ as well? Or we need something more to make this ...
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1answer
29 views
Intuitive way to understand the square wave spectrum?
What is a intuitive way to understand that a transform of a square wave can result into something like this?
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26 views
Decreasing the computational speed of Gaussian elimination of a complex linear system in a special case.
The solution of the complex linear system $Ax = b$ of $n$ equations can be computed using
Gaussian elimination with $O(n^3)$ complex multiplications.
However, how can we show that if ...
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2answers
28 views
Definition of Sobolev Space
I have a definition that says that the space of functions that satisfy$$\|u\|_{H^m}^2=\sum_{k\in\mathbb{Z}}(1+|k|^2)^m|\hat{u}_k|^2<\infty$$is called Sobolev Space and when $m=1$, this is ...
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Dirac Delta or Dirac delta function?
Is Dirac delta a function? What is its contribution to analysis?
What I know about it:
It is infinite at 0 and 0 everywhere else. Its integration is 1 and I know how does it come.
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1answer
77 views
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What will be the support of the convolution of two test functions.
If $g\in C^{\infty}_c$ defined on $\Bbb R^n$ and K is the support of function $g$. I want to find the support of $g_\epsilon$. Where $g_\epsilon$ is regularization of $g$.
Regularization of $g$ is ...
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Fourier analysis questions
Can anyone give me a hand with the proof of this properties?
Prove that:
a) The linear span of the set $\left\{T_bh/b\in\mathbb{R}\right\}$ is dense in $L_2(\mathbb{R})$, where $h(x)=e^{-\pi x^2}$. ...
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Conditions for matrix operator to preserve complex symmetry on DFT vector?
Suppose there is a DFT vector $\mathbf{X}$ (complex vector) with length N, which presents complex conjugate symmetry around its middle point, i.e., $X(1) = X(N-1)^*$, $X(2) = X(N - 2)^*$ and so forth. ...
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Fourier Transforms of shifted sinc funtions
I would like to calculate the Fourier transform of the following functions:
$$\left(\dfrac{\sin(\pi x\pm\pi n/2)}{\pi x\pm\pi n/2}\right)^2$$
$$\dfrac{\sin(\pi x+\pi n/2)}{\pi x+\pi ...
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2answers
34 views
Problem with Discrete Parseval's Theorem
I think I must be missing something obvious, but I can't for the life of me see what it is. The discrete version of Parseval's theorem can be written like this:
$\sum_{n=0}^{N-1} |x[n]|^2 = ...
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1answer
37 views
Fourier analysis question, orthonormal basis.
I need some help with this exercise:
Given $A>0$, let $L_{A}^2(\mathbb{R})$ the subspace of $L^2(\mathbb{R})$ of the functions $f$ that satisfy $\hat{f}=\chi_{[\frac{-A}{2},\frac{A}{2}]}\hat{f}$. ...
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1answer
15 views
Problem with Discrete Parseval's Theorem
I think I must be missing something obvious, but I can't for the life of me see what it is. The discrete version of Parseval's theorem can be written like this:
$\sum_{n=0}^{N-1} |x[n]|^2 = ...
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0answers
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Phase of 2d Fourier transform
I have a 2d function: r(x,y)
I calculated the 2d-Fourier transform of r(x,y): R(wx,wy)
and i want to calculate the 2D-Fourier ...
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Fourier Transform of $(\operatorname{sinc}(x))^2\cdot \exp(-ax^2)$
I would like to calculate the FT of the following function:
$$(\operatorname{sinc}(x))^2\cdot \exp(-ax^2)$$
Any hint is highly appreciated!
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2D Fourier transform of exponentials and cosines
I would like to know the 2D FT of the following functions:
1.$$\exp\left(-\frac{(x-y+a)^2}{b^2}\right)$$
...
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26 views
Square equivalent of $circ(r)$
I would like to know if there is a similar function to
$$circ(\sqrt{x^2+y^2})=1 , 0\leq \sqrt{x^2+y^2}\leq 1$$
but with a square domain $0\leq x\leq 1$ and $0\leq y\leq 1$.
If yes, which is its ...
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2answers
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Complex-valued Fourier integral: $ \int_{ - \infty }^{ + \infty } {\frac{{\cos (ax)}}{{{x^2} + 1}}{e^{ - ibx}}\,\mathrm dx} $
I'm working on the Fourier transform, but I don't know how to evaluate the integral:
$$I = \int_{ - \infty }^{ + \infty } {\frac{{\cos (ax)}}{{{x^2} + 1}}{e^{ - ibx}}\,\mathrm dx} $$
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1answer
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$\int_{-\infty}^{\infty}\!e^{- \pi (x+iy)^2}\,dx = 1$ for all $y$.
Can anyone provide a proof of why $\int _{-\infty} ^ {\infty} e^{-\pi (x+iy)^2} dx$ equals 1, for all y ? $x$ and $y$ are real numbers.
EDIT: We already know this for y=0.
Thank you.
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1answer
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Need help with Fourier transform problem
I'm trying to calculate the Fourier transform of the unit step function,
$$\mathcal{F}[u(t)] \ = \int_{-\infty}^{\infty}u(t)e^{-i\omega t}dt \ = \int_{0}^{\infty}e^{-i\omega t} dt. \tag{1}$$
This ...
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2answers
27 views
Find triple functions $ (g_0,g_1,g_2)$ such that $g_0+g_1'+g_2'' = \delta_0-\delta_1$
I want to find a triple of compactly supported continuous functions $ (g_0,g_1,g_2)$ on $\mathbb{R}$ such that $$g_0+g_1'+g_2'' = \delta_0-\delta_1$$
This is seemingly not so hard but ive broken my ...
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Parseval's identity
How to prove the Parseval's identity , I know the formal way but how to justify the interchange between the integral and the sum in a rigorously way , in addition what extra condition does the ...
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Show that$ a$ is a differential of order $m$.
Lat $a = a(x,\zeta) \in S_{1,0}^m(\mathbb{R}^n,\mathbb{R}^n)$. Write $n=n_1+n_2$ with $n_2\geq 1$ and $\zeta = (\zeta_1,\zeta_2)$ with $\zeta_i\in \mathbb{R}^{n_i}$. Suppose that $a$ does not depend ...
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Fourier analysis exercise
I need a hand with this question:
If $f\in{L_1(\mathbb{R})}$ and $g\in{L_2(\mathbb{R})}$, then prove that $\widehat{f*g}=\hat{f}\cdot \hat{g}$
As a tip, i have been told to prove that:
...
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Fourier transform of integral operator
I would like to know which is the fourier transform of integral operator:
$$Tf(x)=\int_{-\infty}^{+\infty}\quad f(x)dx\rightarrow \hat{T}\hat{f}(k)$$
I know that (is it right?):
...
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Fourier Analysis of a Time Series
The picture below displays experimental data on concentration oscillations in a chemical reaction. I would like to find frequency characteristics of the series.
More specifically, assuming that the ...
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0answers
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Mathematically inclined books on Signal Processing Theory
First off, i know this may seem off topic but i could not find help in signal processing communities so i was hoping there would be people here who both love mathematics and have interest in signal ...
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Extending the multiplication property of the Fourier transform to $L^2(\mathbb{R})$
I've been reading Introduction to Fourier Analysis on Euclidean Spaces by Stein and Weiss and I've come across another item in a proof that I didn't understand. They establish the multiplication ...
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Signed amplitude spectrum
I have never seen a "signed amplitude spectrum" of a Fourier transform in the literature. Let $X(\omega)$ be the Fourier transform, which can be represented as the product ...
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1answer
25 views
Equivalent of definite integral in fourier space
I would like to know the equivalent in Fourier space of an double integral over a circular domain:
$$\int\int_C f(x,y) =\int_0^l \rho d\rho\int_0^{2\pi}d\theta f(\rho,\theta)\rightarrow ????????$$
I ...
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Inverse fourier transform involving exponentials
I would like to calculate the inverse fourier transform of:
$$\hat{f}(k)=\exp(-a k^2+ikv)\cdot \frac{\sinh(m\sqrt{(b+ck^2+ikf)^2-d})}{\sqrt{(b+ck^2+ikf)^2-d}}$$
Any clue?
Thanks!
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Discrete fractional fourier transform
I have written a code for producing matrix of fractional fourier transform with the help of eigen vectors of fourier transfom matrix. Does anyone know the elements of this matrix ( for example a 4 by ...
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1answer
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How to integrate $f(x)$?
I've been asked to integrate $\int{f(x)}^2 dx$ between the ranges of $L$ and $-L$.
I'm stuck! I understand how to integrate a constant or a function as in $x^2$ or something, but the $f(x)$ format is ...
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Intregral of exponential of Shannon Entropy Function
Here I am going to ask a similar question as rde asked , that is what is the integral of exponential of entropy function. That is what is the value of
$F[H(x)]=\int_{-1}^{+1} e^{ikH(f(x^2))} dx$
...
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Verifying Convolution Identities (Not yet Resolved! Help please!)
Note: I don't yet have a solution to my main issue yet which I have elaborated on in the edit. Further attention is deeply appreciated. :>
$\bf{\text{Original Question}}$:
Let $G$ be a locally ...
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A basic question about $\operatorname{supp}f$ (support of f).
Is it true that $\operatorname{supp}f$ is the complement of the biggest open set where $f=0
$?
Here $\operatorname{supp}f=$ {$x\in \Bbb R^n ; f(x)\not=0$} and $f\in C$ (collection of continuous maps ...
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P.d.f of a discrete fourier transform of binary variables
Let $\{a_n\}$ be a set of $N$ "binary" random variables uniformly distribuited in $\{-1,1\}$.
The discrete fourier transform is defined
$b_k=\frac{1}{\sqrt{N}}\sum_{n=0}^{N-1} a_n e^{-2 \pi i k n ...
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1answer
52 views
Numerical Approximation of the Continuous Fourier Transform
Given a function $F(k)$ in frequency space (sufficiently nice enough, eg. a Gaussian), I would like to compute its Fourier inverse
\begin{equation}f(x) = ...
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convolution density of the sum of N random variables
In books it is often stated the convolution that the density of the sum of Xi iid random variables is the convolution:
Assuming i goes from 1 to 4.
We can note the proba density of this sum in ...
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In my Fourier text book, there are the following exercises to prove. why do some of them have the same left side but have different right sides?
In my Fourier text book, there are the following exercises to prove.why do some of them have the same left side but have different right sides? The demand of these question is to prove these ...
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Dense set in $L^2$
Let $ \Omega\subset \mathbb{R}^n$ with $m(\Omega^c)=0 $. Then how can we show that $ \mathcal{F}(C_{0}^{\infty}(\Omega))$ (here $ \mathcal{F}$ denotes the fourier transform) is dense in $L^2$(or ...
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Zero-Padding seems to CHANGE FFT of signal! [migrated]
I am having trouble with zero-padding. As I understand it, in theory, zero-padding in the time-domain should 'sinc-interpolate' the FFT of the signal in the frequency domain.
This would mean, that ...
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Find the magnitude and phase spectra of H(s) = 1/s+1
Find the magnitude and phase spectra of
H(s) = 1 / s+1
I have no clue what this is asking. My teacher kind of left me hanging could someone help me get started.
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183 views
Closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$.
What is the closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$? We can use Fourier series of $e^{-bx}$ ($|x|<\pi$) to evaluate $\sum_{n=-\infty}^{\infty}\frac{1}{n^2+b^2}$. But this ...
