Questions tagged [fourier-transform]

The Fourier transform is important in mathematics, engineering, and the physical sciences. It is a tool that breaks a waveform (a function or signal) into an alternate representation, characterized by sine and cosines. The Fourier Transform shows that any waveform can be re-written as the sum of sinusoidal functions. It is named after French mathematician Joseph Fourier.

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13 views

Correlation between FT of function constisting of a product and FT of its factors

Given is as function $ C(x)=A(x)B(x) $ and I want to find the connection between its fourier transforms. So I want to find a function $$ \tilde{C}(k)=\tilde{C}(\tilde{A}(k),\tilde{B}(k)) $$ ($\tilde{}$...
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A curious derivation of the Gaussian integral from Plancherel's Theorem

I was playing around with the Plancherel's theorem and stumbled across a derivation of the Gaussian integral which I have never seen before. Could folks check to see if derivation is correct? If so, ...
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Finite-duration function in the time domain, means what in the frequency domain?

I'm trying to understand exactly what restrictions a finite duration time domain signal has on its Fourier transform. I found a helpful table in The Fourier Transform and its Applications by Ronald ...
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30 views

Complicated Fourier transform

Let $t \in \mathbb{R}, k \in \mathbb{N}$. Then $$ t^k \widehat{e^{-at} \mathbf{1}_{\mathbb{R}^+}}(t)(\xi) = \frac{k!}{(a+i\xi)^{k+1}} $$ (image) How to show this result ? I tried a IPP in the integral,...
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Fourier transform on two variables

I have got a project to compute numericaly Poisson's and Laplace's equation with spectral method. I have implemented it and now I want to check it's error with analitical approach and here is a ...
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21 views

Show that the solution to the Neumann problem is $u(x,y)=c+\frac{1}{2\pi}\int_{-\infty}^{\infty}{f(\xi)\ln[y^2+(x-\xi)^2]} d\xi$

Show that the solution to the following Neumann problem \begin{cases} u_{xx} + u_{yy} =0 \hspace{0.5cm} -\infty <x<\infty, y>0\\ u_{y}(x,0) = f(x), \hspace{0.5cm} -\infty <x<\infty\\ u(...
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Fourier Transform of $\frac{1}{2\pi} \int\limits_{0}^{2\pi}e^{ir\sin\theta}e^{2i\theta} d\theta$?

The integral relationship for a function $J(r)$ is- $$J(r)=\frac{1}{2\pi} \int\limits_{0}^{2\pi}e^{ir\sin\theta}e^{2i\theta} d\theta$$ How can I determine the Fourier transform of the above function $...
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24 views

Laplace's equation not homogeneous in polar coordinates

Find the solution to the following boundary value problem. \begin{cases} u_{rr} + \frac{1}{r}u_{r}+ u_{zz} = -q(r,z), \hspace{0,5cm} 0<r<1, \hspace{0.2cm}0<z<h\\ u(r,0) = 0, \hspace{0.4cm} ...
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Fourier Transform of Heaviside-like functions (with different $t=0$ values)

Consider two functions $x_1(t)$ and $x_2(t)$ as follows: $$ x_1(t)=\left\{\begin{array}{ll} 0 & t<0 \\ 1 & t \geq 0 \end{array}\right. $$ $$ x_2(t)=\left\{\begin{array}{ll} 0 & t\leq 0 \...
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Finding residue to show reverse Fourier Transform

Show from direct integration that the Fourier transform of the function $ f(t) = \begin{cases} 1, & \text{-1 < t < 0} \\ -1, & \text{0 < t < 1} \\ 0, & \text{otherwise} \\ \...
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Proof for uncertainty (resolution) of Short Time Fourier Transform

I'm looking for a proof for the uncertainty relation of the Short Time Fourier transform (STFT). I would be very pleased if someone has a literature / source recommendation for an understandable proof....
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Fourier transform of $\frac{ds}{dt} = \nabla^2{s^2}$ [closed]

I want to take the spatial Fourier transform of $\frac{ds}{dt} = \nabla^2{s^2}$ Can someone help me with this. I know that the the fourier transform of the del operator multiplies $F[s]$ by $iw$ but ...
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How do we take the spatial fourier transform of this function?

I want to take the spatial fourier transform of the following PDE: $$\frac{\partial \phi }{\partial t} = \nabla^2 [ -\phi (1-\phi^2) - \kappa \nabla^2 \phi].$$ I understand that taking the spatial ...
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Computation of a Fourier transform of a tempered distribution

If $x$ is a Schwartz function and $\theta$ is a tempered distribution given by $$\theta(x)=\int_{0}^{\pi}e^{it}x(t)dt.$$ What is the Fourier transform of $\theta$? I know $\mathcal{F}\theta(x)=\theta\...
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1answer
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Harmonic-analysis Fourier series proof - general fourier series.

Let $f(x) \in L^2[-\pi,\pi]$ such that $f(x)$ suffice: $f(x) \sim \frac{a_0}{\sqrt{2}} + \sum_{n=1}^{\infty}a_n\cos(nx)+b_n\cos(nx) $ $f(x)$ is also even function. Prove: $ \int_{-\pi}^{\pi} f^2(x)\...
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38 views

Write $D^{\alpha} f$ as a convolution

I am trying to figure out an analysis problem related to Fourier transform and Young's convolution inequality. Here is the problem statement: Let $f \in L^1(\mathbb{R}^n) \cap L^p(\mathbb{R}^n) \cap C^...
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27 views

>Prove if the Fourier transform of $f(t)$ is $F(η)$, then the Fourier transform of $F(t)$ is $f(−η)$.

Prove if the Fourier transform of $f(t)$ is $F(η)$, then the Fourier transform of $F(t)$ is $f(−η)$. First I'm not sure what does it mean $F(t)$ because $F(\eta)=\int_{-\infty}^{\infty}f(t)e^{-2\pi t\...
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What is x* when referring to the conjugate symmetry of a Fourier transform?

I'm learning about symmetry properties of the discrete fourier transform. Any sequence x[n] can be expressed as: $x[n] = x_e[n] + x_o[n]$ $x_e[n]$ is a conjugate symmetric component of the sequence ($...
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Solve heat equation with $w(x,0)=\exp(-|x|)$

I've been working on this problem for a while now, for context, I'm allowed to solved using Fourier integral, transform or series, but the most recent topic was the Laplace transform which is the ...
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63 views

Showing that the Fourier transform of a compactly supported Holder function satisfies an estimate

I'm working on a problem that says: If $f \in C_c(\mathbb{R})$ is a compactly supported function that's Holder-continuous of degree $\alpha \in (0, 1)$ (i.e. there exists $C$ such that $|f(y) - f(x)| \...
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What is the one-sided Fourier transform of a constant?

A definition of the Fourier transform commonly used is (I always forget which convention of normalization to use) \begin{align}f(\omega)=\int_{-\infty}^\infty e^{i \omega t}f(t) dt\end{align} For a ...
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Graph convolution using a diagonal filter

Given an undirected graph $\mathcal{G} = (V, E)$, if we consider a signal $\bf{x} \in \mathbb{R}^n$ ($x_i =$ value at node $i$-th) and a filter $\bf{g} \in \mathbb{R}^n$ then we can define a notion of ...
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21 views

Finite integral involving Airy and trigonometric functions

I need to calculate the inverse Fourier transform of the function $$F(w)= N \operatorname{Ai}\left(\frac{|w|-e}{\gamma}\right),$$ where $N$ and $e$ are some constants. The Fourier transform (or ...
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37 views

Fourier transform for the a complex expression

I am stuck with deriving the Fourier transform of an expression mentioned below: $X(t) = \sin(\omega_1 t)\frac{a - e^2 \sin^2( \omega_2 t)}{b - e^2 \sin^2( \omega_2 t)}$ where $a\neq b$ and $a/e^2, b/...
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$f\in L^2(\mathbb{R})$ is absolutely continuous and $f' \in L^2(\mathbb{R})$ if and only if $\int |y\hat{f}(y)|^2dy<\infty$

This comes from Walter Rudin's Functional Analysis p. 388, exercise 21(c): But the domain I found on Engel's One-Parameter Semigroups for Linear Evolution Equations p. 66 is $$ D(A)=\{f\in L^2(\...
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Units of Fast Fourier Transform

I am using Python to apply a fast Fourier transform on a list of solar capacity factors to find the most important periods. I am looking at the amplitudes and periods, but I'm stumped on the amplitude ...
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How do I calculate the inverse Fourier transform of the delta function?

In the context of single-pixel imaging, the following statement is given: "A Fourier basis pattern $P_F (x,y) $ can be obtained by applying an inverse Fourier transform $\delta_F (u, v, \phi)$to ...
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Approximating modified wavenumber

From [1] and [2], the modified wavenumber $\Phi(\vartheta)$ is defined as follow $$\Phi(\vartheta)=-\frac{1}{i\sigma} \ln\left(\hat{\upsilon}(\vartheta;\,\Delta t)\right),$$ where $\sigma=\Delta t/\...
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32 views

Time-shifting and time-scaling of Fourier transform $x(t) = \text{rect} \big(\frac{t}{2}-\frac{3}{2}\big)$

Consider $$x(t) = \text{rect} \left(\frac{t}{2}-\frac{3}{2}\right)$$ From FT-Table, we know $$\mathcal{F}\big(x(at)\big) = \frac{1}{|a|}X\left(j\frac{\omega}{a}\right)$$ and $$\mathcal{F}\big(x(t-t_0)\...
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Can someone explain period finding in either the Discrete or Quantum Fourier Transform (Shor's algorithm)

I am learning about Shor's algorithm, a way to find factors of large numbers using a quantum computer. One of the main steps of this algorithm relies of the Quantum Fourier Transform (QFT) - a quantum ...
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22 views

Calculate this sum based on the Fourier cosine transform of $e^{-x}$

Given that the Fourier cosine transform of $f(x)=e^{-x}$ is: $$f(x)=e^{-x}=\frac{1-e^{-π}}{π}+\frac{2}{π}\sum_{n=1}^\infty\frac{1+e^{-π}(-1)^{n+1}}{n^2+1}\cos({nx}),0\leq x \leq π$$ calculate the ...
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Gelfand transform is onto $C_0(\widehat{L^1(G)})$ but the fourier transform is not onto $C_0(\widehat{G})$

Let $G$ be a locally compact abelian group Denote $\widehat{G}$ the continuous characters of $G$. We equip $\widehat{G}$ the compact-open topology. Denote $\widehat{L^1(G)}$ the set of non-zero ...
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28 views

Changing PDE with scaled and shifted Fourier transform

I have defined a transform ($p,q,c$ are constants) $$V_{p,q}(\omega,\tau,z) = \frac{1}{2\pi}\int e^{-ih(\tau - (p+q)z/c)}U_{p,q}(\omega,h,z)dh$$ so that $$U_{p,q}(\omega,h,z) = \int V_{p,q}(\omega,h,z)...
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56 views

Calculate $\int_{\mathbb{R}}e^{ix\omega}(\frac{\sin(\omega)}{\omega})^2 d\omega$ using convolution theorem.

I have study Fourier transformation and found this exercise: Calculate $\int_{\mathbb{R}}e^{ix\omega}(\frac{\sin(\omega)}{\omega})^2 d\omega$ using convolution theorem. I have notice that integral ...
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17 views

Fourier transform of a generalized complex exponential function

I have the following function $g(k)$, which I would like to perform the inverse Fourier transform upon, i.e., $g(k)\to \tilde{g}(x)$: \begin{align} g(k) \equiv \prod_{n=0}^{\infty} e^{i^n \alpha_n k^n}...
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33 views

Solve $u_{tt} = c^2 u_{xx} + h(x,t)$ using Fourier transform

Consider the problem: $$ \begin{cases} u_{tt} = c^2 u_{xx} + h(x,t), \hspace{0.5cm} x \in \mathbb{R}, \hspace{0.3cm} t>0\\ u(x,0) = f(x), \hspace{0.5cm} x \in \mathbb{R}\\ u_{t}(x,0) = g(x) \hspace{...
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32 views

4th order PDE — solved using Fourier Transforms?

Say I had a PDE: $$u_t + u_{xxxx} + u = f(x)$$ with IC $u(t=0)=s(x)$. How does one use Fourier transforms to solve this? If it were second order ($u_{xx}+u_{yy} = 0$), one could use a sine/cosine ...
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solve the infinite chord problem with the Fourier transform

I was trying solve infinite rope,I need to solve this problem with a different method than the traditional one, (homogeneous case, with D'alambert). Using Fourier transform I think it could be easier ...
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64 views

Fourier transform of $t^n\exp(-\alpha\vert t\vert)$ $(n\geq0)$ [closed]

I met this Fourier transformation when dealing with bath properties in physics. I would appreciate any clue on this.
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Fourier transform of a piecewise function $ f(x)=\begin{cases} x^2,& (-1<x<1),\\ 0,& (\text{Elsewhere}). \end{cases}$

I want to calculate the full-range Fourier transform of the function $f(x)$ defined by $$f(x) = \begin{cases} x^2,& -1<x<1,\\ 0,& \text{Elsewhere}. \end{cases}$$ I calculated it as $$\...
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34 views

Poisson formula with fourier transform

if $f\in C^{1}(\mathbb{R})$ is such that, exists $K>0$ with $|f(x)|+|f'(x)|\leq K(1+x^2)^{-1}$ for all $x\in \mathbb{R}$, then, for all $L>0$: $$\sum_{k=-\infty}^{+\infty} f(x+2kL)=\frac{\sqrt{2\...
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If $\hat{g}(\omega)=e^{-4 a\pi^2 \omega^2 t}$ then how I show $g=\dfrac{1}{4\pi \sqrt{a\pi t}}\exp\Big(\frac{-x^2}{16a\pi^2 t}\Big)$? [duplicate]

In this paper ,( page 3),Author solved heat equation using Fourier transformation method ,such that he come up to $u(x,t)=f(x)*g(t)$ with $$\hat{g}(\omega)=e^{-4 a\pi^2 \omega^2 t}$$ and $f(x)$ is ...
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31 views

4th order PDE — Can I use Fourier transforms?

How does one approach the PDE $$\frac{\partial u}{\partial t} + \frac{\partial^4 u}{\partial x^4} + u = f(x)$$ And solve it using Fourier transforms? I am not clear on the methods of solving 4th order ...
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25 views

Fourier transform in polar coordinates

I have this formula that should be the Fourier transform for a generic coordinate system in the plane: $\vec E(\vec k)=\int \vec E(\vec x) e^{-i \vec k \cdot \vec x} dx^2$ However I don't understand ...
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31 views

Some points concerning the paper entitled “Eigenvalue and Eigen- vector Decomposition of the Discrete Fourier Transform”

I had a look at the interesting paper (see this link) written by JAMES H. McCLELLAN and THOMAS W. PARKS. In Lemma 2, two lists of N-periodic even and odd vectors are given. As for the vector $u_j$, it ...
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20 views

Can the sound of infinitely long played music be decomposed in sine or cosine forms?

I asked this question already on the physics site but it was advised to ask it here. If a piece of music is played for an infinite amount of time (with a volume that you can hear) can it be decomposed ...
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1answer
45 views

Fourier transform of high dimensional PDE

Suppose that $$\partial_t f = \nabla_x \cdot (D\nabla_x f + Cxf),\quad x \in \mathbb{R}^d,$$ in which $D$ is a fixed positive semi-definite matrix and $C$ is a fixed matrix of dimension $d \times d.$ ...
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14 views

Do similar characteristic functions imply quantitative similarity in CDF?

Suppose two $d$-dimensional r.v. $\xi,\eta$ have similar characteristic functions $\varphi_\xi(t)$ and $\varphi_\eta(t)$, like $$\sup_{t\in \mathbb R^d} |\varphi_\xi(t) - \varphi_\eta(t)| \leq \...
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1answer
40 views

The Fourier Coefficients of $\cos(x)$

To begin I simply went with the definition: $$f(t)=\sum_{n=-\infty}^{\infty} f_k\cdot e^{ikw_ot} \implies f_k=\frac{1}{T}\int_{0}^{T} f(t)\cdot e^{-ikw_ot} \,dt$$ And before plugging $\cos(x)$ I first ...

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