Questions tagged [fourier-transform]

For question related to Fourier transforms.

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calculating inverse fourier transform of $\cosh(t\sqrt{1-k^2})$

So i am trying to calculate the following integral \begin{equation} \int_{-\infty}^{\infty}e^{-ikx}\cosh(t\sqrt{1-k^2})dk \end{equation} I think this should be related to modified Bessel function of ...
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Question regarding the indexes $\alpha$ and $\beta$ in the definition of Schwartz space

I was asked to prove that the Fourier transform of any Schwartz function belongs to the Schwartz class, i.e: $$||\xi^{\alpha}\frac{d^{|\beta|}\hat{f}}{d\xi^{\beta}}||<\infty$$ For $\alpha$ and $\...
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The definition of Fourier Integral seems wrong in my textbook.

The definition of Fourier Integral seems wrong in my textbook. The definition in book goes like this, If $f(x)$ is continuous and $\int_{-l}^{l}\left | f(x) \right |dx$ converges to zero as $l$ ...
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Distributions whose Fourier transforms have discrete and countable support (follow-up to finite support)

This is a follow-up on my last question about distributions, Distributions whose Fourier transforms have finite support, again to potentially help with figuring out the solutions for the post ...
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Proving the existence of sine and cosine Fourier transforms

I am working through the following exercise: Let $f\colon \mathbb{R} \rightarrow \mathbb{R}$ be a Riemann integrable function on every $[a,b],$ and such that $\int_{-\infty}^\infty |f(x)|\mathrm{d}x &...
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Computing the Fourier transform of a general disk

Let $\text{circ}(x, y; R, c)$ be the characteristic function on the plane of a disk of radius $R$ and centre $c=(c_1,c_2)$: $$\text{circ}\left(\frac{\sqrt{(x_1-c_1)^2+(x_2-c_2)^2}}{R}\right)=\begin{...
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Fourier transform of hyperbolic functions

I am looking for the Fourier transform of $x \tanh(x)$ in $\mathbb{R}$. The Fourier transform on $\mathbb{R}\setminus\{0\}$ is $$2\sum_{n=0}^{\infty}(2n+1)\exp\left(-(2n+1)\frac{|t|}{2}\right).$$ But ...
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Equality holds in Donoho-Stark iff $f$ is the characteristic function of the translate of a subgroup (finite abelian group)

Let $G$ be a finite abelian group, $f:G\to \mathbb C$ not identically zero. Donoho-Stark uncertainty principle say that $|\textrm{supp} f||\textrm{supp} \hat{f}|\geq |G|$. The prove I have goes as ...
confusedTurtle's user avatar
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Decay property of the Fourier transform

Let $f,g:\mathbb{R} \to (0,\infty)$ be nonnegative functions on $\mathbb{R}$. If $$ \lim_{|x|\to \infty} \frac{g(x)}{f(x)} = 0. $$ Can we conclude anything about the decay property of their respective ...
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Spherical fourier transform [closed]

$$\psi_{2, 1, 0}(r, \theta, \phi) = \frac{1}{\sqrt{6}} \left(\frac{r}{a_0}\right) e^{-\frac{r}{2a_0}} \cdot \sqrt{\frac{3}{4\pi}} \cos(\theta) $$ I want to express the above wavefunction in momentum ...
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Fourier transform $\left[\mathrm{csch}(x+i\epsilon-t)\right]^n\left[\mathrm{csch}(x+i\epsilon+t)\right]^m$

In a physics related problem, I am trying to compute the Fourier transform \begin{align} \mathcal{F}\left[\frac{1}{\sinh^{n}\left[\pi T_R\left(x+ i\epsilon-t\right)\right]\sinh^{m}\left[\pi T_L\left(...
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Trouble calculating Fourier Transform of Dirac's Delta function

I've seen some other posts were already posted on this matter, but none of them seem to be oriented to performing the actual fourier transform of $\delta(x)$. I've just been introduced the Fourier ...
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$ \frac{\pi}{6} \int_{-\pi}^{\pi}e^{-jx\sin(\tau)}d\tau + \sum_{m=-\infty}^{+\infty}\frac{1}{2\pi m^2}\int_{-\pi}^{\pi}e^{2m\tau-x\sin(\tau)}d\tau $

Introduction I'm grappling with an expression that intriguingly combines integrals and series, involving exponential functions with sinusoidal inputs. I'm curious about expressing this in terms of a ...
Alireza Ghazavi's user avatar
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Fourier transform of the product between Heaviside and multivariate Gaussian

Let $0\neq\mathbf{w}\in\mathbb{R}^n$ and denote by $H:\mathbb{R}\rightarrow\mathbb{R}$ the Heaviside step function. I need to calculate the Fourier transform of: $f(\mathbf{x})=H(\langle \mathbf{w},\...
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Computing a fourier transform using FFT

Say I wish to compute the expression $\hat{f}(k)=\frac{1}{L}\int_L f(x)e^{ikx}dx$ using a FFT, so a discrete version of this would be $\hat{f}(k)=\sum f(x)e^{ikx}$ over modes of $k$ defined by $L$. ...
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Fourier transform on a regular lattice: The restricted set of wave-vectors

I) Consider a function defined only on the vertices of a 1D regular lattice: $f_i \equiv f(x_i)$ for all $x_i$, $i \in \{ 1, 2, ..., N \}$ and $x_{i+1} - x_i = a$, where $a > 0$ is the “lattice ...
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Fourier coefficients

I am a computer scientist and not a mathematician. I do not know how to edit questions on this site, so I am rephrasing a problem I came across while reading on the explicit construction of expander ...
james dunk's user avatar
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Fourier transform of a constant from finite to infinite [duplicate]

The inverse Fourier transform of a constant is defined as $\delta(t)$. However, if I use the definition and consider this as a limit \begin{equation} r(t)= \int_{-B}^{B} e^{j\omega t} d\omega = \frac{...
Xiangyu Meng's user avatar
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An additional condition to prove the Fourier transform is bounded

Recently, I read Methods of Modern Mathematical Physics by M. Reed and B. Simon to learn Fourier transform, there is a lemma before proving Fourier inversion theorem: The maps $\hat{f}(\vec\lambda)=\...
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Dispersive relation from modified equation Beam-Warming

I'm looking at this example regarding dispersive relations from modified PDE equations: I'm working with $a=1$. I derived the modified equation, but I don't understand how the author makes the ...
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Is the Lomb-Scargle periodogram the same as Schuster periodogram for aperiodically sampled complex data?

I found a paper by Brethorst where he developed a periodogram that is a generalized version of the Lomb-Scargle periodogram. You can find it here. I tried to implement (22) from this paper to make a ...
CfourPiO's user avatar
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Oscillatory Integrals near the Riemann singularity

The question comes from E. M. Stein. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, we concerned about the highly oscillatory distribution $$ D(x)=\mathrm{p.v.} \...
InnocentFive's user avatar
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Fourier transform on Schwartz space leads to identity.

Let $f$ be a function in the Schwartz space and $F(f) = \left(\frac{1}{2 \pi}\right)^{n/2} \int_{R^n} e^{i \langle x,y \rangle } f(x) dx $ the Fourier transform. Is it true that $F^4(f) = f$? I know ...
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Question about computing the Fourier transform of a product

[After further searching around, I decided to modify the question slightly] Let $A$ be an invertible real matrix and $g(\mathbf{x}) = e^{ i \mathbf{x}^T A \mathbf{x} }$. Let $w$ be a real valued ...
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How to solve the Fourier transformation of a function of the form $e^{-(x-a)^2}$

I have no idea how to continue to find a solution for an Fourier transformation of the form $\exp\left\{-(x-a)^2\right\}$, this is what I tried (already found): $$\mathcal{F}\{f\}(\omega)= \int_{-\...
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Eigenvalue problem for linearized reaction-diffusion system

I recently started studying about reaction-diffusion system and Hopf bifurcation theory. I realize that Fourier transform/series is a quite useful tool here but it's been many years since I got ...
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Fourier transformation with a square root-log term

(Note: I posted the exact same question in the physics StackExchange, but to get a breadth of people looking at the problem, I am coming to the Math StackExchange also since, well, it is just an ...
MathZilla's user avatar
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Integration of discontinuous functions

In order to evaluate its Fourier transform, I want to determine whether $f(x)=\arctan(\frac{1}{x})$ belongs in $L^1(\mathbb{R})$, $L^2(\mathbb{R})$ or both. Therefore, we have to check the continuity ...
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How does the Fourier transform map a function into a distribution?

So far I understand how the Fourier transform maps Schwartz space into Schwartz space, $L^2$ to $L^2$, and how it can be extended to map tempered distributions to tempered distributions. However one ...
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function and its second derivative in $L^2(\mathbb{R})$ implies $f^\prime$ in $L^2$.

Suppose $f$ is smooth and both $f$ and $f^{\prime\prime}$ are in $L^2(\mathbb{R})$. Prove $f^{\prime}$ is again in $L^2(\mathbb{R})$. Someone said I can use the method of Fourier transform. Because ...
Xinlin Wu's user avatar
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How does the exponential in the Fourier Transform affect its accuracy?

I know that the length of the time interval of a function affects the accuracy of its Fourier Transform. So when we take a longer interval of a function, it is easier to distinguish between the ...
Aya Noaman's user avatar
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Is the more general uncertainty principle related to bias-variance tradeoff?

Since I first saw this video on the more general uncertainty principle, I have taken the bias-variance tradeoff to be an example of it. They seem quite similar on the surface. However, now that I'm ...
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Understanding the Fourier transform of $f(x; \alpha) = \frac{1}{|x|\left(\mathrm{ln}|x|\right)^\alpha}$

Let $\alpha > 0$. I'm looking to better understand the Fourier transform of the function $$f(x; \alpha) = \frac{1}{|x|\left(\mathrm{ln}|x|\right)^\alpha}$$ In particular, I would like to know ...
Cartesian Bear's user avatar
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Fourier Transform of an impulse function

We have to find the Fourier transform of the following impulse response $$ h(t)=T[\delta (t)]=\frac{1}{2t_{d}}(\delta (t+t_{d})-\delta (t-t_{d})) $$ My own approach led to the following: $$ H(\omega )=...
Yurian -'s user avatar
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Smoothness of Fourier transform

I am trying to understand, because the Fourier transform of the function $f(x) = e^{ -\sqrt{ \lvert x \rvert } }$ is smooth. My question: Under which conditions is the Fourier transform of an $L^1$ or ...
TheFibonacciEffect's user avatar
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Showing that Sobolev Space $H^m$ is in $L^\infty$

I'm very new to Fourier analysis/Sobolev spaces and am stuck on this exercise. I found proofs of more general embedding theorems for Sobolev spaces and some similar questions on here, but they are too ...
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Calculating the Integral: $\int_{-\infty}^{\infty} e^{j(w_0-w)t} dt$ [duplicate]

I have encountered the following integral: $$\int_{-\infty}^{\infty} e^{j(w_0-w)t} dt$$ where $w_0$ and $w$ are constants. I know that the result of this integral is $2\pi\delta(w-w_0)$, where $\delta$...
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Fourier transform of $g(x - \frac{xy}{f}, y)$

Let $g(x,y)$ have Fourier transform $G(k_x, k_y)$. I am interested in the Fourier transform of $g(x - \frac{xy}{f}, y)$ for some $f > 0$. I've gotten part of the way there using the separability ...
Vivek's user avatar
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Why isn't the fourier transform defined differently?

I never really understood why the amplitude of the Fourier transform does not represent the amplitude of a sin/cos signal of a specific frequency? The way the Fourier transform is typically defined, ...
LEXOR AI's user avatar
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Approximating the surface measure of spheres via Fourier transform

Recently I'm reading a paper by Magyar-Stein-Wainger, Discrete analogues in harmonic analysis:Spherical averages, Annals of Mathematics, 155 (2002), 189–208. A key ingredient in the proof of their ...
Tutukeainie's user avatar
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Fourier Transform Duality Theorem scaling problem?

From the duality theorem if: $$ FT(x(t)) = X(w) $$ Then $$ FT(X(t)) = 2\pi x(-w) $$ If I choose $X(t) = 1$, then $X(w) = 1$ thus $x(t) = \delta(t)$. Now plugging back into the second equation: $$ FT(1)...
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Form of fourier series in 3D coordinate

I have an interesting problem where a constrained solver is used to estimate a periodic function as part of a sensor calibration process. The idea is to write this periodic function in the form of ...
Anh Tran's user avatar
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Relationship between Fourier inversion theorem and convergence of "nested Fourier series representations" of $f(x)$

In this question I use the term "nested Fourier series representation" to refer to an infinite series of one or more Fourier series versus a single Fourier series. Whereas a single Fourier ...
Steven Clark's user avatar
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Computing the 2D greyscale image produced from putting a row and column of zeros between every two rows and two columns in the Fourier transform $F$

I have the 2 dimensional image $f(x,y)$ of dimensions $K\cdot T$, and there is its Fourier transform $F(u,v)$. Now, I want to compute the image $g(x,y)$ which is of dimensions $(2K)\cdot(2T)$, which ...
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Limit of a Fourier Trasformation

Knowing that $f \in L^2(\mathbb{R^n})$, then $$\lim_{\epsilon\to0+}\int_{\mathbb{R^n}}e^{-i\langle x,\xi\rangle -\epsilon|x|}f (x)dx=\mathcal{F}_2f(\xi)$$ in $L^2(\mathbb{R^n})$, where $\mathcal{F}_2f$...
Andreadel1988's user avatar
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Fourier transform and weak solution

I have the following problem $$u_t-\Delta u=f,$$ for $(x,t)\in \mathbb{R}^n\times (0,\infty)$ and $$u(x,0)=u_0(x)$$ for $x\in\mathbb{R}^n$. The function $f\in L^2(0,T;L^2(\mathbb{R}^n))$.I have to ...
Gonzalo de Ulloa's user avatar
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1 answer
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Can the inverse Fourier transform $f(x)=\mathcal{F}_{\omega}^{-1}[F(\omega)](x)$ converge when $F(\omega)=\mathcal{F}_x[f(x)](\omega)$ doesn't?

Are there examples of functions $f(x)$ for which the inverse Fourier transform integral $$f(x)=\mathcal{F}_{\omega}^{-1}[F(\omega)](x)=\int\limits_{-\infty}^\infty F(\omega)\, e^{2 \pi i x \omega}\, d\...
Steven Clark's user avatar
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Question about derivation of Fourier transform

I'm reading the book "Fourier Analysis and Its Applications" and in the derivation of the Fourier transform he began with writing the Fourier series $$f(t)=\sum_{n=-\infty }^{\infty }c_{n}...
Mans's user avatar
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Fourier series of surface in terms of spherical harmonics

During my studies I came across the following integral $$\oint_S\mathrm{d}S\,\mathrm{e}^{-\mathrm{i}\vec{x}\cdot\vec{k}},$$ where $S$ is a 2D surface in 3D space, and noticed that it almost looks like ...
Caesar.tcl's user avatar
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The correct way of looking at Fourier transform

I Know that Fourier transform states that any non-periodic function could be described as summation of sines and cosines by saying that $$F(w)=\int_{-\infty }^{\infty }f(x)e^{^{-iwt}}dt$$ And this was ...
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