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 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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1answer
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Fourier transform over a diagonal matrix
Let $F$ be a $100 \times 100$ DFT matrix, and $U$ be a diagonal matrix with its diagonal entries being all positive, denoted by $U=\mathrm{diag}(u_1, u_2,\cdots, u_{100})$. My question is:
Under ...
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2answers
25 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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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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1answer
42 views
Fourier Transforms of shifted sinc funtions
I would like to calculate the Fourier transform of the following functions:
$$\left(\dfrac{\sin(\pi x\pm\pi n/2)}{\pi x\pm\pi n/2}\right)^2$$
$$\dfrac{\sin(\pi x+\pi n/2)}{\pi x+\pi ...
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3answers
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Dirac Delta or Dirac delta function?
Is Dirac delta a function? What is its contribution to analysis?
What I know about it:
It is infinite at 0 and 0 everywhere else. Its integration is 1 and I know how does it come.
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1answer
29 views
Confusion of one definition in Fourier analysis
The symbol occurs on Page 22 of Bahouri's book Fourier analysis and nonlinear differential equations.
As defined there,
$$f(D)a:=\mathcal{F}^{-1}\{f\mathcal{F}a\}.$$
The question comes from the ...
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27 views
Support of the convolution of two test funtions.
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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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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1answer
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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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2answers
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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
35 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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Is $\{g(n)\}_{n\in\mathbb{Z}}\in\ell_2$ if $g$ is a Sobolev function on the real line?
If we are given a function $g\in W_2^k(\mathbb{R})$ (even consider $k=1$ for simplicity), then is it true or not that $\{g(n)\}_{n\in\mathbb{Z}}\in\ell_2$? That is, do we have ...
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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)$$
Any hint is highly appreciated!
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0answers
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Phase of 2d Fourier transform
I have a 2d function: r(x,y)
I calculated the 2d-Fourier transform of r(x,y): R(wx,wy)
and i want to calculate the 2D-Fourier ...
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2D Fourier transform of exponentials and cosines
I would like to know the 2D FT of the following functions:
1.$$\exp\left(-\frac{(x-y+a)^2}{b^2}\right)$$
...
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1answer
59 views
Fourier analysis exercise
I need a hand with this question:
If $f\in{L_1(\mathbb{R})}$ and $g\in{L_2(\mathbb{R})}$, then prove that $\widehat{f*g}=\hat{f}\cdot \hat{g}$
As a tip, i have been told to prove that:
...
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25 views
Square equivalent of $circ(r)$
I would like to know if there is a similar function to
$$circ(\sqrt{x^2+y^2})=1 , 0\leq \sqrt{x^2+y^2}\leq 1$$
but with a square domain $0\leq x\leq 1$ and $0\leq y\leq 1$.
If yes, which is its ...
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1answer
56 views
$\int_{-\infty}^{\infty}\!e^{- \pi (x+iy)^2}\,dx = 1$ for all $y$.
Can anyone provide a proof of why $\int _{-\infty} ^ {\infty} e^{-\pi (x+iy)^2} dx$ equals 1, for all y ? $x$ and $y$ are real numbers.
EDIT: We already know this for y=0.
Thank you.
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1answer
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Need help with Fourier transform problem
I'm trying to calculate the Fourier transform of the unit step function,
$$\mathcal{F}[u(t)] \ = \int_{-\infty}^{\infty}u(t)e^{-i\omega t}dt \ = \int_{0}^{\infty}e^{-i\omega t} dt. \tag{1}$$
This ...
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2answers
495 views
Advice for how to learn more advanced math for audio signal processing?
I am very interested in learning about audio from a signal processing standpoint. However, whenever I try to further my education by reading books, I get extremely frustrated because the books use ...
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2answers
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Find triple functions $ (g_0,g_1,g_2)$ such that $g_0+g_1'+g_2'' = \delta_0-\delta_1$
I want to find a triple of compactly supported continuous functions $ (g_0,g_1,g_2)$ on $\mathbb{R}$ such that $$g_0+g_1'+g_2'' = \delta_0-\delta_1$$
This is seemingly not so hard but ive broken my ...
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1answer
49 views
Help proving Calderón reproducing formula (simple version)
Let $\phi$ be a real compactly supported smooth function on $\mathbb R$ with total integral zero. Define $\phi_t=\frac{1}{t} \phi(\frac{x}{t})$. I also suspect that they must be even, but the notes I ...
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1answer
31 views
Parseval's identity
How to prove the Parseval's identity , I know the formal way but how to justify the interchange between the integral and the sum in a rigorously way , in addition what extra condition does the ...
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Show that$ a$ is a differential of order $m$.
Lat $a = a(x,\zeta) \in S_{1,0}^m(\mathbb{R}^n,\mathbb{R}^n)$. Write $n=n_1+n_2$ with $n_2\geq 1$ and $\zeta = (\zeta_1,\zeta_2)$ with $\zeta_i\in \mathbb{R}^{n_i}$. Suppose that $a$ does not depend ...
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1answer
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Fourier transform of integral operator
I would like to know which is the fourier transform of integral operator:
$$Tf(x)=\int_{-\infty}^{+\infty}\quad f(x)dx\rightarrow \hat{T}\hat{f}(k)$$
I know that (is it right?):
...
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0answers
21 views
Fourier Analysis of a Time Series
The picture below displays experimental data on concentration oscillations in a chemical reaction. I would like to find frequency characteristics of the series.
More specifically, assuming that the ...
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Mathematically inclined books on Signal Processing Theory
First off, i know this may seem off topic but i could not find help in signal processing communities so i was hoping there would be people here who both love mathematics and have interest in signal ...
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Signed amplitude spectrum
I have never seen a "signed amplitude spectrum" of a Fourier transform in the literature. Let $X(\omega)$ be the Fourier transform, which can be represented as the product ...
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Extending the multiplication property of the Fourier transform to $L^2(\mathbb{R})$
I've been reading Introduction to Fourier Analysis on Euclidean Spaces by Stein and Weiss and I've come across another item in a proof that I didn't understand. They establish the multiplication ...
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1answer
25 views
Equivalent of definite integral in fourier space
I would like to know the equivalent in Fourier space of an double integral over a circular domain:
$$\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}}$$
Any clue?
Thanks!
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1answer
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Verifying Convolution Identities (Not yet Resolved! Help please!)
Note: I don't yet have a solution to my main issue yet which I have elaborated on in the edit. Further attention is deeply appreciated. :>
$\bf{\text{Original Question}}$:
Let $G$ be a locally ...
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1answer
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How to integrate $f(x)$?
I've been asked to integrate $\int{f(x)}^2 dx$ between the ranges of $L$ and $-L$.
I'm stuck! I understand how to integrate a constant or a function as in $x^2$ or something, but the $f(x)$ format is ...
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Discrete fractional fourier transform
I have written a code for producing matrix of fractional fourier transform with the help of eigen vectors of fourier transfom matrix. Does anyone know the elements of this matrix ( for example a 4 by ...
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1answer
239 views
Show that f is a polynomial
Suppose $f$ is an entire function on $\mathbb{C}^n$ that satisfies for every $\epsilon>0$ a growth-condition $$|f(z)|\leq C_{\epsilon}(1+|z|)^{N_{\epsilon}}e^{\epsilon |
\text{Im}\,z|}$$
...
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Intregral of exponential of Shannon Entropy Function
Here I am going to ask a similar question as rde asked , that is what is the integral of exponential of entropy function. That is what is the value of
$F[H(x)]=\int_{-1}^{+1} e^{ikH(f(x^2))} dx$
...
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1answer
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Inverse Fourier Transforms in 3D
How do you calculate the inverse Fourier transform
http://www.fuw.edu.pl/~mklis/publications/Hydro/oseen.pdf
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4answers
272 views
Singular asymptotics of Gaussian integrals with periodic perturbations
At the bottom of page 5 of this paper by Giedrius Alkauskas it is claimed that, for a $1$-periodic continuous function $f$,
$$
\int_{-\infty}^{\infty} f(x) e^{-Ax^2}\,dx = \sqrt{\frac{\pi}{A}} ...
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1answer
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Estimate derivatives in terms of derivatives of the Fourier transform.
Let us suppose that $f: \mathbb{R}^n \to \mathbb{R}$ is a smooth function. Furthermore, for every $\alpha$ multi-index, there exists $C_\alpha > 0$ such that
$$
|D^\alpha f(\xi)| \leq ...
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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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1answer
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Proving that a certain function is in $W^{1,n}(B(0,1))$
Fix $0<\alpha<1-\frac{1}{n}$ and let $f\colon\mathbb{R}^n \rightarrow \mathbb{R}$ be the function $f(x)=(\log(\frac{1}{|x|}))^{\alpha}$.
How can I prove that $f\in W^{1,n}(B(0,1))$?
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1answer
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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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1answer
162 views
Real Analysis Qualifying Exam Problem
I think this should be an easy question, and I believe the answer should be in the positive, but I am not sure how to start. I would appreciate some help. Thank you.
Suppose that $f_j$ is a ...
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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
48 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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1answer
39 views
number of zeros of the superposition/interference of sine oscillations
There is a tricky problem to solve and we ask for your kind help.
In a rather simple version of the problem, we have a set of oscillations of type $f(x)=\sin (\omega x + \phi)$ with different phases ...
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2answers
81 views
Show existence of a continuous $k$ on $\mathbb{R}^2$ such that $(u,\phi)= \int_{\mathbb{R}^2}k\phi dx$ for all $\phi$
(b). Let $u$ be a distribution on $\mathbb{R}^2$. Assume there exists a continuous function $h$ on $\mathbb{R}^2$ such that $(u,\Delta \phi) = \int_{\mathbb{R}^2}h\phi dx $ for all $\phi\in ...
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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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1answer
68 views
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 ...
