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Questions tagged [fourier-transform]

continuous Fourier transform, discrete Fourier transform (DFT), discrete cosine transform (DCT), discrete sine transform (DST)

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Partial Differential Equation using Fourier Sine Transform

Hello everyone I was doing a bit of homework when I ran into the following problem: PDE: $$u_t= \alpha^2 u_{xx}, \text{for } 0\le x \lt \infty$$ With boundary condition of: BC: $$u_x(0,t) = 0, \text{...
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Fourier transform of a function with singularity

Anybdoy knows where to find the Fourier transform of the function $$\frac{1}{|x|^\alpha}$$ where $0<\alpha<N$ and N is the dimension. I onu know this for schwarz space. Anyone can say to me ...
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Understanding/calculating the fourier coefficients of multiplied functions

I am hoping to get some clarifications/help on dealing with coefficients of a multi-dimensional Fourier series. First, I apologize for any mistakes or notations that may be off, I know just enough ...
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Separation of variables for nonhomogeneous equations

I’m trying to learn PDE from An introduction to partial differential equations, Pinchover and Rubinstein. On page 114 section 5.4 it explains the use of separation of variables for nonhomogeneous ...
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Derivative of the Forward Fourier Transform

I'm trying to differentiate the forward fourier transform with respect to $k$, I ended up with $$-i \int_{-\infty}^{\infty} x f(x) e^{-ikx}dx $$ How do I simplify this further, then generalise it ...
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A Fourier transform calculation

I can't figure out how to derive equation A.4 from this paper, Simon, Barry. "Some Jacobi matrices with decaying potential and dense point spectrum." Communications in Mathematical Physics 87, no. 2 (...
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51 views

Using the result $g(\omega)=\frac{1}{\sqrt{2\pi}}\frac{2\alpha}{\alpha^2 + \omega^2}$ evaluate $\int_{-\infty}^{\infty}\frac{1}{1+x^2}e^{-ix}dx$

For this question the following definitions of the Fourier transform and its inverse are used: $$g(\omega)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(t)e^{i \omega t}dt\tag{1}$$ $$f(t)=\frac{1}{\...
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How to evaluate the imaginary part of this one-sided fourier transform?

So, I came across the following integral $\tag{1}\Gamma(\omega) = \int_{0}^{\infty}dse^{i\omega s}G^{+}(s)$ where $G^{+}(s) = \langle \phi(t)\phi(t - s)\rangle = \left[-16\pi\alpha^2\sinh^2(\frac{s}...
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Fast Fourier transform of the Gaussian function

As I know, the Fourier transform of the Gaussian function is also Gaussian. The simple verification for that is shown below. http://mathworld.wolfram.com/FourierTransformGaussian.html But fast ...
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Fourier Transform of $\exp(-at)(1-\cos(wt))$. [on hold]

may I ask if one can derive the Fourier Transform of the following? $e^{-at}(1-\cos(bt))$ Thanks is advance
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Solve this PDE using Fourier Transform

Question: Solve the following system using Fourier Transform: \begin{alignat}{2} & \nabla^2 f = 0 & z<0 \\ & \frac{\partial f}{\partial z} = \frac{\partial g}{\partial x} \quad \frac{\...
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35 views

Inverse Fourier transform of complex fourier transform

How can i find the inverse fourier transform when the fourier transform is given in complex form the module $$|x(\omega)|=|\omega|$$ and the $$\phi(\omega)=-3\omega$$ should i find the inverse ...
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Interchanging Integration Order involving Fourier Transform

$$f(\omega,u):=\frac1{\omega+iu}$$ where $i$ is the imaginary unit number. We see that the integral of a Fourier transform $$\int_1^\infty du\int_{-\infty}^\infty d\omega\,f(\omega,u)\,e^{-i\omega x}=...
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How can I get transformed coordinate between different basis?

For example, there is set of DCT basis which are orthogonal: $F_1(x,y),F_2(x,y)\cdots,F_N(x,y)$. So, given function can be uniquely expressed as sum of DCT basis. (we decompose an image to sum of DCT ...
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1answer
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Fourier transform, same frequencies, different amplitudes

I understand that the Fourier transform is changing the domain (time/space) to frequency domain and provides the sin waves. I have seen the visualizations of Fourier transform and they are all showing ...
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PDE: Find an explicit expression for the solution of the IVP by using Fourier Transform

Find an explicit expression for the solution of the IVP: $u_{t}+cu_{x}+u=0$ $u(0,x)=f(x)$ by using Fourier Transform That's how I started but don't quite understand what to do after! Any help ...
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How to transform basis functions

I know how to transform the basis if we have two sets of basis vectors. Now in my situation, I have two basis equation and I want to find out the transform between those basis functions. Specifically, ...
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Fourier Transform of Dyson's Equation

I'm attempting to show that the Fourier transform of Dyson's equation for a constant potential V, \begin{equation} G(\mathbf{r},t,\mathbf{r}_0,t_0) = G^0(\mathbf{r},t,\...
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A clarification of Fourier representation of a function

As I was studying Fourier representation of a $T$-periodic function $f(t)$, I came up with two different definitions. For a function with discrete domain, $f(t) = \sum\limits_{k=0}^{T-1} C_k e^{i2\pi ...
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Number of linearly independent solutions after Fourier transform of an ODE

For linear ODEs the number of linearly independent solutions is equal to the order of the equation. However, it's not clear to me how to recover the same number of solutions in I'm using Fourier ...
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Does this definition of the Fourier intensity measure make sense?

Let $\epsilon_n$ be a sequence in $\{-1,1\}^{\mathbb Z_+}$. EDIT: to make the question easier, assume that $\epsilon_n$ is just the Thue-Morse sequence with symbols $1$ and $-1$. We define partial ...
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Find $f\in L^1(\mathbb{R})$ such that $\|\int_{-N}^N \hat{f}(\xi)e^{2\pi i\xi \cdot}\operatorname{d}\xi-f\|_1\nrightarrow 0, N\to\infty?$

I'm wondering how to find a function $f\in L^1(\mathbb{R})$ such that, if $\hat{f}$ is its Fourier transform, i.e.: $$\forall \xi\in\mathbb{R},\hat f (\xi):=\int_\mathbb{R} f(t)e^{-2\pi i\xi t}\...
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1answer
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Interpretation of $\tilde{f}(\mathbf{0})$

Given a function in real space $f(\mathbf{r})$, what is the interpretation of the of the value $\tilde{f}{(\mathbf{0})}$? As an example, take the Fourier transform of \begin{equation} V(\mathbf{r}) ...
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Confused on Fourier Transform

Currently I am reading a paper related to Computer Graphics. However, the core algorithm in the paper uses Fourier Transform. I learned Fourier Transform by myself and I still don't understand some ...
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1answer
49 views

Fourier Transform of a Spherical Well Potential - Rotating the System

I'm attempting to take the Fourier transform of the following function: \begin{equation} V(\mathbf{r}) = \begin{cases} V_0 & r<r_0 \\ ...
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How to re-scale and do correctly the discrete Fourier transform.

I'm stuck with the re-scaling and the proper choice of parameters in doing the discrete Fourier transform. I explain: Suppose you want to calculate the Fourier transform $$ F(p) = \frac{1}{2\pi}\int ...
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1answer
49 views

Integral of Complex Gaussian: $\int_{-\infty}^{\infty} e^{-(2\pi x +i\omega)^2}dx$.

I wonder if the integral $\int_{-\infty}^{{\infty}}e^{-\alpha x^2}=\sqrt{\frac{\pi}{\alpha}}$, for $\alpha\neq 0$, how could the integral $\int_{-\infty}^{\infty} e^{-(2\pi x +i\omega)^2}dx$ be ...
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Fourier transform of $e^{i\sqrt{1+x^2}}$

As the title says: I want to compute the Fourier transform (in the distributional sense) of $f(x)=e^{i\sqrt{1+x^2}}$, $x\in \mathbb{R}^n$ - say $n=1$ for the moment. I have no idea how to get it done: ...
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eigenfunction representation with spline: show that coefficients fall faster than order of eigenvalues

I'm trying to understand the Proof of Theorem 3 in Bühlmann & Yu 2003 (Boosting with the $L_2$-Loss). The paper considers some projection matrix $S$ corresponding to a smoothing spline of degree $...
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Fourier Transform of the Riemann zeros (Dirac comb)?

Lets assume RH and $\rho_i, i\in\Bbb N$ be the imaginary parts of the non-trivial zeros of the Riemann $\zeta$ function: $\zeta(\frac{1}{2}\pm\imath \rho_i)=0$, $(\forall i)$. Does anonye know if ...
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Fourier transform on Schwartz Space is a continous automorphism

I am currently trying to understand the Fouriertransform of Schwartz functions. There are two proofs I'm trying to figure out. $\mathcal{F}$ is a continous operator $\mathcal{S}(\mathbb R) \...
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Fourier representation of a function with changing period

I learned that the Fourier representation of a periodic function $f(t)$ with period $T$ is $\sum \limits_{k=-\infty}^{\infty} C_k e^{i2\pi k t/T}$ with appropriate constants $\{C_k\}$. My question: ...
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Fourier support of a real function

For a real function it is known that $$ F(\xi_1,\xi_2) = F^*(-\xi_1, -\xi_2), $$ but does it also imply that $$ F(\xi_1,\xi_2) = F^*(+\xi_1, -\xi_2), $$ i.e., that the transform should be symmetric in ...
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1answer
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Convolution - references request

From which book this chapter is ? http://www.math.ncku.edu.tw/~rchen/2016%20Teaching/Chapter%203_Convolution.pdf
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A naive and easy formula to calculate convolution power

In mathematics (in particular, functional analysis) convolution is a mathematical operation on two functions ($f$ and $g$) to produce a third function that expresses how the shape of one is modified ...
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Hankel transformation and inverse Hankel transformation of integer order.

Read the definition of Hankel transformation here... https://en.m.wikipedia.org/wiki/Hankel_transform Question: can we define Hankel transformation for any Integer order? Can we define inverse Hankel ...
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Integral involving a Gaussian and a rational function.

Let $a \ge 0$ and $b \ge 0$ be real numbers. By generalizing the approach from Evaluating $\int_{\mathbb{R}}\frac{\exp(-x^2)}{1+x^2}\,\mathrm{d}x$ . we have derived the following results. Let $n\ge ...
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Computing the Fourier Transform of a Discrete Shearlet

Let $f\in L^2(\mathbb{R}^2)$ we denote it´s Fourier transform by $\hat{f}(\xi)=\int_{\mathbb{R}^2}f(x)e^{-2\pi i\langle x,\xi\rangle}dx$, where $\langle \cdot,\cdot\rangle$ is the inner product on $\...
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1answer
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Heat equation PDE (nonhomogeneous); Green's function; Dirac delta

(Sorry for the messy title, trying to include the keypoints of the problem.) I am new to the theory on how to solve this kind of PDE problem which is presented below; I am unsure on which method to ...
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1answer
24 views

Fourier transform of an absolute value

I have this function: $$f(t)=\left | \cos\Bigl(\frac{2\pi t }{T}\Bigr) \right |$$, for $$\left | t \right |\leq \frac{5T}{4}$$ and zero otherwise. I have to find its Fourier transform and plot its ...
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Discrete Fourier Transform of Sinc Function

We know that the discrete Fourier transform (DFT) of a discrete rectangular function is related to Dirichlet kernel: $D_n$(x)=$\frac{sin[(n+1/2)x]}{sin(x/2)}$, and the Fourier transform of a ...
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Poisson as a limit of the Binomial Characteristic Function

We are given $X_n\sim B(n,p_n)$ where $np_n\rightarrow\lambda$, and $\lambda>0$. The goal is to prove $X_n$ converges in distribution to Poisson($\lambda$) by use of characteristic functions. ...
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Fourier transform of $\theta(x-L)f(x)$?

Is there anything special one can say about the fourier transform $$\hat g(y)=\frac{1}{2\pi}\int_{-\infty}^\infty dx e^{i x y} g(x)$$ of the function $$g(x)=\theta(x-L)f(x)$$ where $L>0$, $\...
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Approximating Fourier transformed function with Fourier series

Given a function $f(x)$, composed with continuous frequencies in certain interval $[-m,m]$,\begin{equation} f(x) = \frac{1}{a}\int_{-m}^{+m} \hat{f}\Big(\frac{\xi}{a}\Big)e^{\frac{i2\pi x\xi} {a}}d\xi;...
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Fourier transform of a windowed cosine function

I was doing some problems for practice and I came across this: $$f(t) = \cos(\omega_0t) \left[H\left(t+\frac{T}{2}\right) - H\left(t- \frac{T}{2}\right)\right]$$ where $H(t)$ is the Heaviside/ ...
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Why are patterns repeated in the frequency-power graph of a periodic signal?

I have a signal, which I'd like to treat as a non-continuous function now, let it be $signal(t)$. It looks like this: Zoomed in a bit: I create a Lomb-Scargle Periodogram using $signal(t)$, which I ...
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Matrix factorisation of the Fourier matrix

I am currently reading a paper Low Communication FMM-Accelerated FFT on GPUs In that I am not able to understand the definition of the twiddle factor matrix $T_{P, M}$. The Fourier matrix $F_N$ is ...
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1answer
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Calculating an integral using a Fourier transform

We have $$f(t)=\frac{1}{t^2+6t+13}\tag{1} \qquad \& \qquad \hat{f}(\omega)=\frac{\pi}{2}\cdot e^{3\omega i}e^{2|\omega|}$$ whereas $\hat{f}$ denotes the Fourier transform. We want to calculate ...
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Is the inverse Fourier transform always defined?

The Fourier transform of a continuous-time function is defined if the function is absolutely integrable, otherwise it does not exist. What about the inverse Fourier transform? If I make up any ...
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energy density spectrum vs energy spectral density

I am doing a project on ocean wave simulation and there is a formula I am trying to test. It is called the random coefficient scheme and it is meant to simulate a random time series. One part of the ...