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

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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23 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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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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15 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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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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28 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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29 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
20 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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38 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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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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13 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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16 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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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 ...
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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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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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47 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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30 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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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
40 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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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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33 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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11 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 ...
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26 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 ...
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1answer
41 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
15 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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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 ...
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2answers
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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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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} ...
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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 ...
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39 views

Solution of boundary value problem using Fourier series

I want to solve the following PDE using Fourier series. $u(x,y): \Omega \to \mathbb{R}$, $\Omega=(0,\pi)\times (0,2\pi)$ $u-3u_{xx}-u_{yy}= 3\sin(2x)-\sin(5x)$ $u_{xx}$ and $u_{yy}$ are second ...
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Explanation of this integral

$$\int_{-\infty}^{\infty} e^{-\frac{i}{\hbar}(p-p')x} dx = 2\pi\hbar\delta(p-p')$$ I don't quite understand how this integration leads to the right hand side. Any explanation is appreciated.
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Example of multidimensional Fourier transform

Please, take the function, for example $\sin(xy)+\cos(xz)$ from dimension 3 to $R$, and give me a multidimensional Fourier transform for it. I'll be also thankful for general multidimensional Fourier ...
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1answer
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Solving integral equation with Fourier transform?

I'm trying to solve the following integral equation using Fourier transforms: $$u(t)+ \int_{-\infty}^{t} e^{\tau-t} u(\tau)\,d\tau=e^{-2|\tau|}$$ I tried to transform both sides of the equation using ...
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1answer
35 views

How to properly shift in frequency domain an already shifted function in time domain?

I would like to shift in frequency domain the following function: $f(t)=\frac{1}{\sqrt{2\pi}\sigma}\exp(-\frac{(t-t_0)^2}{2\sigma^2})$. As usual, frequency shift will introduce a new term ...
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2answers
47 views

Convolution Integral to Evaluate Fourier Transform

According to Mathematica with Fourier transform convention $$\widehat{f}(\xi)=(2\pi)^{-1/2}\int_{-\infty}^{\infty}f(x)e^{i\pi x}dx$$ The Fourier transform of the function $f(x):=|x|^{-1/2}e^{-|x|}$ ...
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When studying 2D gabor functions why is a gaussian called elliptical?

Consider $$G(x,y)=\frac{1}{2\pi\sigma\beta}e^{-\pi\left[\frac{(x-x_0)^2}{\sigma^2}+\frac{(y-y_0)^2}{\beta^2}\right]}e^{i[\xi_0x+\nu_0y]}.$$ This is the product of a complex plane wave and what this ...
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What happens with Fourier transform of a Gaussian function?

I am facing the following problem: I have a Gaussian function of time with the following form $f(t)=\frac{1}{\sqrt{2\pi}\sigma}\exp(-\frac{t^2}{2\sigma_2})$. I would like to make a shift both in time ...
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37 views

Reference request: about inverse Laplacian operator

I am currently studying some problems about inverse Laplacian and the Yosida approximation and wishing to learn more about it. Here is a post about one of the problems that I am interested in. ...
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21 views

Does this transformation(harmonic analysis) exist?

Assuming that there are two disjoint sets(A and B) of high dimensional $N^d$ integers. Each point $V$ can be expressed by a periodic function: $f(t)=v_1*cos(t) + v_2*sin(2t) + ...
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17 views

How to identify a process via its Karhunen-Loeve expansion?

Suppose that you are given the following Karhunen-Loève expansion of a real-valued continuous Gaussian stochastic process, $x$. $$x(t) = \sum_{k=1}^{\infty}z_{k}\cdot \frac{\sqrt{2}\sin((k-0.5)\pi ...
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1answer
43 views

fourier transform of $f(x) = x^2+\frac{1}{1+2x^4}$

I really have no thought on this. I can't seem to use residue thm., nor could I find a inverse transform for it. by some Fourier Calculator I know it's solvable but how?
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2answers
21 views

Harmonic Motion - Fourier Approximation What does this mean below?

There is a method to solve systems under harmonic loading, harmonic balance method, which is basically obtaining fourier expansions of unknown response quantities and solving for coefficients of ...