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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Imaginary number and absolute value integral - Fourier transform

I came across this integral problem: $$\hat f(\xi)=\int_{-\infty}^{+\infty} e^{-|x|+xi\xi}dx$$ Now I know how to integrate simple absolute value functions like: $\int_{-2}^{4}|x-2| dx$, we just ...
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16 views

Fourier transform on forced linear harmonic oscillator

I'm having trouble solving the following equation: $\ddot x + \omega^2 x = a \sin \Omega t$, where $t >0$, $\omega \neq \Omega$, $x(0+) = 0 = \dot x(0+)$. It is asked to be done by Fourier ...
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9 views

Discrete Fourier Transform DFT

I'm having trouble locating a satisfactory reference on discrete Fourier transform (DFT). I would like to get a clear understanding for 1. DFT over $\mathbb{C}$ and 2. DFT over a finite field ...
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1answer
29 views

How to use the 3rd and 4th boundary conditions in this?

I was solving $$ \frac{\partial^2 u}{\partial t^2}=\frac{\partial^2 u}{\partial x^2}$$ All the boundary conditions are as follows:- $$u(0,t)=0 \\ u(\pi ,t)=0 \\ u(x,0)=\sin x \\ u_t(x,0)=x^2$$ ...
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4answers
63 views

Why cannot $A\sin\alpha x +B\cos \alpha x$ be zero?

I was going through solving wave equations using fourier and I came across a note saying $A\sin\alpha x +B\cos \alpha x \neq 0$ I believe this applies to $\alpha ,A,B\neq 0$ I was solving $$ ...
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3 views

Fractional Sobolev space on a compact 1-D segment

Fractional Sobolev space $H^s_p(\mathbb R), s>0, 1<p<\infty$ is a space of tempered distributions $f$ that satisfy $F^{-1}((1+|\xi|^2)^{s/2} F(f)) \in L^p(\mathbb R)$. Here, $F$ denotes the ...
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14 views

Help in plotting the Z-transform and Fourier Transform of the following Sequences.

I'm taking Digital Signal Processing class at the moment and while I believe I understand the theory behind the z-transform and fourier transform, in this case DFT and DTFT, I'm stuck as to how to ...
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8 views

Why is there periodicity in the output of Richard Voss' fractional Brownian motion?

I am trying to figure out why the output of fractional Brownian motion (fBm) as described by Richard Voss (Random fractal forgeries. In: Fundamental Algorithms for Computer Graphics, R. A. Earnshaw ...
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1answer
73 views

How can I show that one of $m(A)$ or $m(\Bbb{R}\setminus A)$ is zero?

Let $A \subseteq \Bbb{R}$ be Borel measurable, and $T$ a dense subset of $\Bbb{R}$. Suppose for every $t \in T$ that $$m((A+t)\setminus A)=0,$$ where $m$ is the Lebesgue measure. Then I want to show ...
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22 views

Fourier limits integral

I´m trying to set up the limits of the Fourier Integral of this periodic function. My doubt is if the interval is from $-\infty$ to $\infty$ or from $0$ to $\infty$.
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21 views

Fourier Transform of derivative a vector function

I am working on the proof of the Fourier Transform of the derivative of a function. I am accompanied by some proof lines but having some issue in one of the integral evaluation. I searched out its ...
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41 views

Exercise 16 in Chapter 5 of Stein and Shakarchi's Real analysis

I'm having trouble with the following problem in Stein's Real Analysis book: Suppose $n$ is the smallest integer $>\frac{d}{2}$. If $$ f \in L^2\mathbb(R^d)$$ and $$(\frac{\partial}{\partial ...
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15 views

Find second moment in (Ito) stochastic differential equation

I'm having a look at a document where a Ito stochastic differential equation is described (to put you in some context, this arises from the equations of motion of a particle embedded in a gas, ...
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2answers
30 views

Fourier transform problem with symmetric matrix. Related to Gaussian?

Hi everyone I encountered a problem that looks simple enough but I have no idea where to start. Find Fourier transform of $e^{-\langle Ax,x\rangle}$ when $A$ is a positive definite symmetric $n ...
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3answers
88 views

Integral of $\int_{\mathbb{R}}e^{-\frac{x^{2}}{2}}\left(\cos\left(\pi nx\right)\right)dx$ [duplicate]

I was in need to urgently solve this integral. I already know the result in the closed form, does anybody know how to solve it? \begin{equation} ...
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1answer
35 views

Odd and Even Fourier Series Extension of $f(x)=x$ on $[0,\pi]$

I'm confused on finding the odd and even extensions of $f(x) = x$ on $[0,\pi]$. I know the general forms and how to find the co-efficients, but for the sin series, $f(0)$ =/= $f(\pi)$, so then I only ...
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1answer
90 views

Solution of this definite integral?

I want to find the expression for the following integral $$\int_0^\infty\text{d}x\frac{e^{i k x}}{x}$$ I have tried deriving with respect to $k$, transforming into an integral over the whole real ...
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17 views

Why is multiplication in frequency domain equals convolution in time domain? [on hold]

The title says it all in my mind. I do not know what else to add. Instead of simply putting the question on hold, could you be so kind as to actually point out what's missing?
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20 views

Function with both easy to find Fourier and Hermitian coefficient

I'm writing some notes on Spectral theory and I would like to make a simple example finding the generalized fourier coefficient of a function in respect of two different bases. I was thinking about ...
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34 views

Fourier coefficients of the Gaussian.

I would need to find the fourier coefficient of this gaussian for a problem. I'm now stuck with this integral, \begin{equation} c_{n}=\int_{-1}^{1}e^{\frac{x^{2}}{2}}\left(\cos\left(\pi ...
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0answers
30 views

Fourier transform calculus of tempered distributions

For example I wanted to ask confirmation of this calculation, if $u \in \mathcal{S}'(\mathbb{R}^n)$ then $\widehat{D^\alpha u} =(2\pi i \xi)^\alpha \widehat{u}$. By definition $\langle \varphi , u ...
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1answer
24 views

Apply Periodic Boundary to PDE (Fourier Transform)

Use Fourier Transform to solve the BVP: \begin{cases} u_t + a u_x - b u_{xx} = 0, & \mbox{for } x \in [-1,1] \\ u(x,0) = f(x) \\ u(x+2,t) = u(x,t) \end{cases} I solved the problem (attached); ...
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41 views

Solving PDE's: Laplace equation on semi-infinite cylinder

Solve Laplace’s equation $\nabla^2$u = 0 inside a semi-infinite cylinder 0 < z, 0 < r < 1 with boundary conditions u(r = 1, $\theta$, z) = $e^{−z}$ and u(r, $\theta$, z = 0) = 0, where (r, ...
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19 views

the ideal structure of group $C^*$-algebras

What is the ideal structure of group $C^*$-algebras? Do there exist any books or articles in the field ? If G to be the group of integers $Z$ , then $C^*$($Z$)=C($T$). so because ideal structure of ...
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14 views

Source for Space-Time Fourier transform theory

I need to do research on Space-Time Fourier transforms (specifically applications within EM theory). Since the resources for learning this digitally seem to be limited (by my very advanced Google ...
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0answers
35 views

Strichartz Estimate with Fourier Transform

Let $f$ be a Schwartz function. Prove that, whenever $2\le r < \infty,$ $$\| e^{it \Delta} f\|_{L^{3r}(\mathbb{R}^2_{xt})} \le c \| \widehat{f}\|_{r'},$$ Where $1/r + 1/r' = 1.$ My Attempt My ...
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0answers
5 views

Biorthogonal (discrete wavelet) noise bases?

I am slightly interested in discrete wavelet transforms (DWT), but so far I have mostly used already-derived and existing well-known wavelets, such as Daubechies, Cohen-Daubechies-Faveau, Symlets and ...
1
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2answers
51 views

How to get this result of integral?

Statement \begin{equation} \int_{\mathbb R} \exp \left( -2\pi (\frac{x}{\sqrt{2}})^2 \right) \exp\left( -i2 \pi \frac {x}{\sqrt{2}} \cdot f \right) dx = \exp \left( -\pi f^{2} / 2 \right) ...
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23 views

Fourier transform of $H(-t)e^{5t}$

i have to calculate the Fourier transform in the title. My professor says the result is $\frac{1}{5-2\pi i f}$. I start from $H(t)e^{\alpha t}$, and i calculate the transform $H(t)e^{5t}\rightarrow ...
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1answer
16 views

Can you kindly explain me in detail this Fourier transform?

I've this function to transform not using the general formula, but just substituting the known transform (i.e. $\text{rect}(t)\rightarrow \text{sinc}(f)$): $\frac{\sin(6\pi t)}{t}$ I know the ...
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68 views

What (if any) are the restrictions on the Fourier transform of a function of the form $f(x) = e^{i \phi(x)}$, where $\phi(x)$ is a real function?

Here I'm defining my Fourier transform according to the convention $$ FT\left\{f(x)\right\} = \hat{f}(q) = \int_{-\infty}^{+\infty}dx\, f(x)\, e^{-i q x}\, . $$ My intuition - which may be off - ...
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31 views

Fourier Transform of $f : x \mapsto \frac{24(5-5x-30x^2-10x^3+5x^4+x^5)}{(2+x+x^2)^5}$

I need of the Fourier Transform of $f$, but I can't solve this. I try to use the integrate and Fourier proprietates but no sucessufull. Help me please $$\ f(x) ...
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1answer
18 views

Tempered representatives of a special class of distributions

Suppose that a distribution $R\in D'(\Bbb R)$ satisfies the following estimation for an independent constant $c$: $$\forall \phi\in D(\Bbb R)\quad |\langle R,\phi\rangle|\le c\|\phi, \,L^1(\Bbb ...
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34 views

Calculating Fast Fourier Transform from given set of data

I am trying to calculate the Fast Fourier Transform numerically from the given data : Given: f0 f1 f2 f3 f4 f5 f6 f7 1 2 3 4 4 3 2 1 I have to find the ...
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1answer
41 views

Inverse Fourier Transform by using convolution theorem.

Inverse Fourier Transform of: $$\mathfrak{F}^{-1} \left \{ e^{-\frac{x^2}{2}}{\frac{sinx}{x}} \right \} $$ by using convolution theorem. Since Fourier Transform convolution turns into ...
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1answer
48 views

How to use Fourier's transform to solve differential equation

I have to solve following problem: $$ u_t(t,x) = \Delta u(t,x) $$ $$ u(0,x) = f(x) $$ I've started: $$ \frac{\delta}{\delta t} F(u(t,\xi))=F(u_t(t,\xi))=F(\Delta u(t,x))$$ and here I've stoped, ...
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15 views

Mathematical Expression for a Fourier Transform $s(T)$

$S(f)$ is the Fourier transform of a non-periodic signal, $s(t)$. $S(f)$ is given by: $S(f) = 1,$ for $−1/2 ≤ f ≤ 1/2$ and $0$ otherwise. What would be a mathematical expression for $s(t)$?
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1answer
34 views

Fourier Transform of $f(t)=4te^{-t^2}$

I am trying to find Fourier Transform of: $$f(t)=4te^{-t^2}$$. I found in MatLab that $\mathfrak{F}\left \{ f(t) \right \}=i\sqrt{2}e^{- \frac{w^2}{4}}w$ .So is this possible to come to same result ...
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1answer
28 views

Solve problem using Fourier's transform

I have a few problems which need to be solved using Fourier's transform. My problem is that I don't know how should I start this type of exercise (I just begin learning differentional equations). ...
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14 views

How do you take the discrete Fourier transform (DFT) of a parallelogram or a Bravais lattice in general?

I'm working on implementing a method that extracts the corresponding wallpaper group given a gray-scale image/pattern. But to do so, I need to take the DFT of a unit cell in the image which, in the ...
2
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2answers
27 views

Use the Fourier transform to find value of definite integral from negative infinity to infinity

Find the value of $\int_{-\infty}^{\infty} f(x) dx$, where $f(x)=sin(x)/(x^3+x)$. How do I go about solving this? I have tried to expand the sine part into complex exponentials to try and resemble ...
2
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1answer
42 views

Fourier transform of $\frac{\sin(6\pi t)}{t}$

I have to calculate the fourier transform of this function in time domain: $\frac{\sin(6\pi t)}{t}$. First I tough to use the definition of $\operatorname{sinc}$ function as ...
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0answers
16 views

Fourier transform of windowed complex exponential

I have a function on the form $$f(x) = g'(x)*e^{i\pi g(x)}.$$ Where $g'(x)$ is a window function with support in the range $-R \ldots R$. I want to find the fourier transform $\mathcal F(\omega)$ ...
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58 views

Fourier distribution $\frac{e^{i|x|}}{|x|}$

I need help to calculate Fourier transform in distribution sense of $\frac{e^{i|x|}}{|x|}$ in $D'(\mathbb{R}^3)$ we have $ \frac{e^{i|x|}}{|x|} \in L^1_{loc}(\mathbb{R}^3)$ edit, Let ...
2
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2answers
40 views

Question about proof of Fourier transform of derivative

If $f\in L^1(\mathbb{R})$, $f'(x)$ exists and is continuous, and $f'\in L^1(\mathbb{R})$, then $\widehat{f'}(t)=2\pi i \widehat{f}(t)$. I've stated the above theorem from a textbook that I'm ...
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17 views

What is an angle in fractional fourier transforms?

I would like to know the geometrical interpretation of an angle in fractional Fourier transforms. Is this a rotation of time-frequency plane or rotation of the signal?
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1answer
25 views

Extension of Fourier transform to complex analytic function

Let $f(x) \in L^1(\Bbb{R})$ have compact support, say $\operatorname{supp}(f) = [-R,R]$. We have the Fourier transform $$\hat{f}(\xi) := \int_{\Bbb{R}} e^{-ix\xi} f(x) dx = \int_{-R}^R e^{-ix \xi} ...
2
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23 views

fourier transform for pde equation

I was solving the pde using fourier transform: $u_{tt}-u_{xx}+m^2u=0$ with initial values $u(0,x)=f(x)$ and $u_t(0,x)=g(x)$. I have received the answer $$U(t,k)=Ae^{-it \sqrt {k^2+m^2}}+Be^{it \sqrt ...
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22 views

Multiplier operators on anisotropic weighted $L^2$ spaces

Suppose $\mathcal{M}$ is a multiplier operator on $L^2(\mathbb{R})$, in the sense that, for any $u(x)\in L^2(\mathbb{R})$, $\widehat{\mathcal{M}u}(k)=m(k)\hat{u}(k),$ where the scalar complex function ...
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21 views

When are Fourier spaces included in each other?

Given the Fourier spaces $V(N_1, T_1)$ and $V(N_2,T_2)$, what necessary and sufficient conditions are required in order to have $V(N_1, T_1)\subset V(N_2,T_2)$? I know that if $V(N_1, T_1)$ is ...