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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To What Extent is the Fourier Inversion Theorem Due to the Self-Adjointedness of the Laplacian

I've tried looking this up (I looked at various spectral theorems) but couldn't find anything that talks about the connection between Fourier transforms and the eigenfunctions of the Laplacian (we may ...
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26 views

How to interpret the imaginary part of an inverse fourier transform

The fourier transform of arbitrary real data can (usually will) result in complex data. If the real data represents samples in time, then the complex FT data represent frequencies with the magnitudes ...
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70 views

$\widehat{f\ast g}= \hat{f} \cdot \hat{g}$ for $f, \hat{f} \in L^{p}(\mathbb R)\cap C(\mathbb R) (1<p<\infty, p\neq 2), g\in \mathcal{S}(\mathbb R)$?

It is well-known that, for $f,g \in L^{1}(\mathbb R).$ Then, by Fubini's theorem, one can derive, $\widehat{f\ast g} = \hat{f} \cdot \hat{g},$ (that is, Fourier transform takes, convolution to point ...
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24 views

Let $f(x)=(1+|x|^2)^{-a}$, with $x\in \Bbb R^n$ and $a>0$.

Let $f(x)=(1+|x|^2)^{-a}$, with $x\in \Bbb R^n$ and $a>0$. Show that $f(x)$ is a constant multiple of $g(x)$. Let $f(x)=(1+|x|^2)^{-a}$, with $x\in \Bbb R^n$ and $a>0$. Show that $f(x)$ is a ...
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20 views

Use the Fourier inversion formula to compute $e^{-a|x|^2}*e^{-b|x|^2}$

Use the Fourier inversion formula to compute $e^{-a|x|^2}*e^{-b|x|^2}$ Let $u=e^{-a|x|^2}$ and $v=e^{-b|x|^2}$. So $u*v=\breve{\hat{u*v}}=(2 \pi)^{-n}\hat{\hat{u*v}}(-x)=(2 ...
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11 views

Use the Fourier inversion formula to compute $\chi_{(-a,a)}*\chi_{(-b,b)}$ where $a,b>0$

Use the Fourier inversion formula to compute $\chi_{(-a,a)}*\chi_{(-b,b)}$ where $a,b>0$. Let $u=\chi_{(-a,a)}(x)$ and $v=\chi_{(-b,b)}(x).$ So, $$u*v=\breve{(\hat{u}\hat{v})}$$ So I calculated ...
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25 views

Show that the Fourier transform of f is in $L^p(R)$ for every $2 \leq p \leq \infty$.

Let $$f(x)=\sum_{n=1}^\infty \sqrt{n} \chi_{(\frac{1}{n+1},\frac{1}{n})}(x)$$. The Fourier transformation of f is $$\hat{f}(y)=\sum_{n=1}^\infty ...
3
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70 views

asymptotical behavior of integral

I'm interest in the asymptotical of $$\int_{-\pi}^{\pi}\exp\Big((\cos z+i\alpha\sin z-1)t\Big)dz\hspace{3mm}\text{as}\hspace{2mm}t\to\infty$$ for $-1<\alpha<1$. Numberical result suggest that ...
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30 views

Is this a Fourier transformation?

As working on a crystallographic project, I saw somebody claimed that they can implement FFT on the following formula: $$F(q)=\sum_rf(r,q)\exp(iqr)$$ How could they do FFT if $f$ is function of both ...
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29 views

Fourier transform of a derivative

My PDE book gives the properties for the fourier transform of $u(x,t)$: $F(\frac{\partial^n}{\partial t^n}u(x,t))=\frac{\partial^n}{\partial t^n}\hat u(\xi,t)$, and $F(\frac{\partial^n}{\partial ...
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13 views

Get a set of points from Fourier plot on Matlab?

I have a program that collects raw data and also does a Fourier transform to get complex numbers. When I feed the raw data into Matlab to do its own FFT, I can get a nice plot of power on frequency. ...
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69 views

What is difference between Fourier Transform and Fast Fourier Transform?

If you think about Fourier Transform, in the classical cases, say on the real line, what it is? Just a waded sum. Right? You take a function $f$, and you take it's Fourier Transform at particular ...
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33 views

Find $\lim_{k \to \infty} f_k$ where the limit is in distribution sense and

Find $\lim_{k \to \infty} f_k$ where the limit is in distribution sense and $f_k(x)=k(1-e^{x/k})\delta_{1/k}$ $f_k(x)=\sin(2 \pi x)\chi_{(k,k+1)}(x)$ When $f_k(x)=k(1-e^{x/k})\delta_{1/k}$ ...
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48 views

Prove that $f_a$ doesn't depend on a.

Let $a>0$ and $$\langle f_a,\psi\rangle=\int_{|x|>a} \frac{\psi(x)}{|x|} dx + \int_{|x|<a} \frac{\psi(x)-\psi(0)}{|x|}dx$$ Prove that $f_a$ does not depend on a. Proof: To prove that $f_a$ ...
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10 views

Partial derivative in frequency domain when only time domain function is known

I want to calculate $$ \frac{\partial F_p(X(\omega))}{\partial X(\omega)} $$ So $F_p$ operates in some way on $X(\omega)$ but I know the analytical form only in time domain, represented by $f_p$. ...
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58 views

Proving that two functions involving integrals with Legendre polynomials are equal

I have two functions that I expect to be equal (where $P_{2l}$ are the even Legendre Polynomials): $$F_{2l}(x)=x\, \tanh(\pi x/2)\left|\int_0^1 u^{i x-1}P_{2l}(u)\,du\right|^2$$ ...
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76 views

Is Fourier transform density preserving?

I know my question is not well-defined since Fourier domain and codomain are not the same, but one knows that they are actually homomorphic. Now what I mean by density preserving is as follows: ...
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45 views

$\int_{\mathbb R^{2}} |\int_{\mathbb R} (f(t-y)- f(t)) g(t-x) e^{-2\pi i w\cdot t} dt|dx dw \to 0 $ as $y\to 0$?

Fact: It is well-known that translation is continuous in the $L^{1}$ norm, that is, if $f\in L^{1}(\mathbb R)$ then $\lim_{y\to 0} \|f_{y}-f\|_{L^{1}(\mathbb R)}=0;$ (where, $f_{y}(x)= f(x-y)$, ...
3
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63 views

Fourier series that converges in $L^2$ but not pointwise

I've read this in my notes Let $f:\mathbb R\to \mathbb R$ be $2\pi$-periodic, and piecewise continuous with jump discontinuities such that $\displaystyle f(a)=\frac{1}{2}\frac{f(a^+)-f(a^-)}{2}$ . ...
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32 views

missing $j*\omega$ in integral

let us consider following integral according to property of delta function,we can write this intgeral as $\int^{t=\infty}_{t=t_0} e^{-j*\omega*t}$ or we can write as ...
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26 views

Show that $u=\sum_{k=0}^\infty k^{-2} \delta_{1/k}$ is a distribution, but …

Show that $u=\sum_{k=0}^\infty k^{-2} \delta_{1/k}$ is a distribution, but $v=\sum_{k=0}^\infty \frac{1}{k} \delta_{1/k}$ is NOT a distribution. Then find the support of u. The definition of ...
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33 views

Solving PDE via Fourier Transform & Uniqueness

When a PDE is solved via Fourier transform, is there already a uniqueness assertion that comes for free? For example, if we Fourier the heat equation \begin{align} \partial_t u(x,t) &= \Delta ...
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14 views

Solution of definite integral of product of bessel function and exponential

I have an integral $I=\int_{\theta} \int_r J_m(k_1r)e^{-j[P_x r \cos(\theta)+P_y r \sin(\theta)]} r dr d\theta$ $0\leq\theta\leq2\pi; r<\infty$ is there any method to solve this?
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43 views

Let $\langle S, \psi \rangle=\sum_{n \in \mathbb N} \int_0^n \psi'(x)dx$. Is S a distribution?

Let $\langle S, \psi\rangle=\sum_{n \in N} \int_0^n \psi'(x)dx$. Is S a distribution? I claim that S is not a distribution. I know that if S was a distribution it would satisfy the following ...
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25 views

Fourier Transform-1

I am trying to solve a Fourier transform problem and I am stuck. The problem is: $$f(t)= \frac{\sin(2t)}{e^{|t|}}.$$ I have used integration, but the answer that I come up with is different than ...
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1answer
29 views

Does there exists $f\in \mathcal{S} (\mathbb R)$ so $\hat{f}=1$ on a comapct set $C$ and $\hat{f}=0$ outside $C\subset W$ (open set)?

Let $C$ is a compact subset of $\mathbb R,$ $V\subset \mathbb R,$ and $0<m(V)<\infty,$ where $m$ is a Lebsgue measure on $\mathbb R.$ My Question is: Can we expect to find $k\in ...
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1answer
40 views

Integrability condition on the Fourier transform implies that the function is infinitely differentiable

This is a problem from the book from Stephane Mallat "A wavelet Tour of signal processing: a sparse way". A function $f$ is bounded and $p$ times continuously differentiable with bounded derivatives ...
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46 views

Fourier-Transformation of Operator

I have an operator $\hat{L}$ which gives $$\hat{L} f(x) = \lambda \cdot f(x)$$ where $\lambda$ is the eigenvalue. Now I Fourier-Transform my function $f(x)$: $$\mathcal{F}(f)(p) = g(p)$$ Question: ...
4
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1answer
63 views

Complex Integration: theorem of Residues

I am trying to prove a theorem that is doing my head in a bit. I have tried to simplify the problem as much as possible and leave out the details, even though it might look a bit too big. The ...
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44 views

Fourier transform help for solving $u_t+u_{xxxx}+u_{xx}=0$

I just started to learn a little bit of fourier analysis in solving PDEs. I want to find a solution $u(x,t)$ to $u_t+u_{xxxx}+u_{xx}=0$. My attempt: Applying the fourier transform to both sides gives ...
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Which of the following functions belong to $S(R^n)=\{ f \epsilon C^\infty(R^n): |x^\alpha| \times |D^\beta f(x)| \leq C_{\alpha, \beta} \}$

Which of the following functions belong to $S(R)=\{ f \epsilon C^\infty(R): |x^\alpha| \times |D^\beta f(x)| \leq C_{\alpha, \beta} \}$? a) $f(x)=\frac{sin(x)}{x}$ b) $f(x)=1-e^{-x^{-2}}$ with ...
4
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1answer
192 views

For every $a>0$, show that $\langle f_a, \psi\rangle= \int_{|x|>a} \frac{\psi(x)}{|x|}dx+\int_{|x|>a} \frac{\psi(x)- \psi(0)}{|x|}dx$

To prove that the inner product is a distribution it must satisfy the following property" $$|T(\phi)|=|\langle T,\psi\rangle| \leq C_N \sum_{|\alpha| \leq N} \|\partial^\alpha \psi\|_\infty$$ Part ...
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40 views

Covariance between real and imaginary parts of Fourier transform of a stationary time series

Since Fourier transform of a random stationary time series(in the case of existence) is not necessarily real, my question is what is the relation between the covariance of real and imaginary parts of ...
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1answer
41 views

$\lim_{k \rightarrow \infty} k(1+\cos(2k \pi x)) \chi_{(-1/2k,1/2k)}$

How do I calculate the following limit: $$\lim_{k \rightarrow \infty} g_k(x) =\lim_{k \rightarrow \infty} k(1+\cos(2k \pi x)) \chi_{\left(\frac{-1}{2k},\frac{1}{2k}\right)}$$
2
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1answer
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fourier transform of scaled function

let us consider following example one thing which i did not understand is where absolute value of $a$ came from?ok if we have $\int^{\infty}_{-\infty} x(a*t)*e^{-j\omega*t}dt$ then we may have ...
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14 views

radon transformation backprojection

I am working on image recontruction and I try to find out how the radon transformation works. I have benn using mainly Natterer, F. and Wubbeling, F.: Mathematical Methods in Image ...
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53 views

Fourier transform of a sinusoidal function

Let us consider following table which I want to calculate myself $$ x(t)=\frac{\sin(\omega_bt)}{\pi t}\quad\iff\quad X(j\omega)= \begin{cases} 1 & \text{if $|\omega|<\omega_b$}, ...
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Fourier series using summation methods

My question is similar to this one. There are ways of deriving the formulae like $$\sum_{k = 1}^\infty \frac{\sin(kz)}{k} = \frac{\pi - z}{2}$$ using summation methods. My question is: How can we ...
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31 views

Filter on Fourier Series

i have a lowpass filter H(ω) which is $ H(ω) = e^{-jω} $ on -2π≤ω≤2π, and $0$ elsewhere and i have a function in fourier series y(t), i need to find the new signal (z(t)) after the application of the ...
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7 views

Inter-neighbor resistance on triangular prism

Given a triangular prism of infinite length along the X direction. A graph is formed with the set of nodes all the points on an edge of the prism with integer values of X, and the with each node ...
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41 views

wavelets, fourier-transform [duplicate]

We are given that $f\in C^{p} \ if \int_\mathbb{R}|\hat{f}(\omega)|(1+|\omega|^p)d\omega <+\infty$. Now if $\hat{f}$ has a compact support then how $f\in C^{\infty}?$
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Fourier transform of a 2-D Gaussian on a ring

I need some help obtaining the 2-D Fourier transform of the following function: $$f(r)=e^{-\frac{-2(r-a)^{2}}{w^{2}}}$$ Where $r$ is the polar radius, $a$ and $w$ are positive. So this describes a ...
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10 views

Spectral norm of a Hadamard product

Let $F$ be the $n\times n$ DFT matrix, i.e., $F_{i,j} = \exp\left(-\tfrac{2\pi(i-1)(j-1)}{n}\imath\right)$, for $1\le i,j\le n$ and $\imath =\sqrt{-1}$. Futhermore, let $\Vert \cdot\Vert$ and $\circ$ ...
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1answer
17 views

How can apply the $L^p$ norm in a circle to $L^2$ norm in a square?

Let $f(x_1,x_2)=\frac{1}{(x_1+ix_2)^2} \chi_{ \{ (-1,1) \times (-1,1) \}}$ Part A: Evaluate the $L^p(R^2)$ norm of $f(x_1,x_2)\chi_{ \{ (x_1,x_2): x_1^2+x_2^2 \leq 1 \}}$ for every $1 \leq p \leq ...
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19 views

Fourier Transform for option pricing

Can Fourier transforms be used to derive the joint probability density function of stochastic interest rates and sotck price Brownian motions of call options under stochastic interest rates? So lets ...
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2answers
39 views

Why Fourier transform owns two different signs?

In my book, the defination of Fourier transform is $$F(\lambda)=\int_{-\infty}^{+\infty}f(t)e^{i\lambda t}dt$$ While the reverse one is: ...
0
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1answer
26 views

Use the interpolation theorem to estimate the $L^p$ norm of f(x) when $p>2$.

The maximum of the function $\displaystyle f(x)=\frac{\sin(x)}{x}$ is $1$ and $\displaystyle \int_{-\infty}^\infty(\frac{\sin(x)}{x})^2 dx= \pi$. Use the interpolation theorem to estimate the $L^p$ ...
2
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1answer
70 views

Noncommutative Fourier Transform

The theory of Fourier transform for Euclidean spaces has analogues for locally compact abelian groups. In the noncommutative setting, representations can be used to define analogous transforms. My ...
0
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1answer
28 views

Discrete Time Fourier Transform of a real signal

I want to prove that if we have a real signal x[n] then for the DTFT it is applied that we have an even symmetry: | X(Ω+1/10) | = | X(-(Ω+1/10)) | (I mean the ...
5
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2answers
52 views

Computation of the fourier transformation of a function with a matrix

I want to compute the Fourier transformation of the following function: \begin{align} f:& \mathbb R^n \rightarrow \mathbb R \\ & x \mapsto \exp(-\left<Ax,x\right>) \end{align} where ...