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Questions tagged [fourier-transform]

For question related to Fourier transforms.

The Fourier transform of a function $f(t)$ is denoted by $\hat f(x)$ and is defined as

$$\hat f(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} f(t)e^{-2\pi itx} dt$$

The inverse Fourier transform is defined as

$$f(t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \hat f(x)e^{2\pi itx} dx$$

4781 questions
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Computing Fourier transform of power law

I'm trying to compute the Fourier transform of $$f(\mathbf{r}) = \frac{1}{r^\alpha}$$ where $\mathbf{r} \in \mathbb{R}^n$. For sufficiently large $\alpha$, the Fourier transform exists. One well-known example in physics is the case $\alpha = n-2$,…
knzhou
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Fourier transform as exponential of Hermitian operator

The Fourier transform $$F: L^2\rightarrow L^2 \qquad \hat{f}(\omega) \equiv (Ff)(\omega) \equiv \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(t)e^{-i\omega t}dt$$ can be viewed as a unitary operator with inverse (and hence, adjoint) $$(F^{-1}Ff)(t)…
poare
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Verify my understanding of a math joke.

I am trying to understand this comic: Note that after hovering over the image more text is revealed namely, "...spike in the Fourier Transform at the one month mark where...". Is the joke referring to an implied jump discontinuity where perhaps…
Natalie Lauf
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Fourier inverse transform of (w-ia/w-ib)

I would like to calculate the Fourier's inverse transform of the following function $$F(\omega)=\frac{\omega-ia}{\omega-ib}, \ \ \ \ \ \ (1)$$ where a and b are real and positive. Hence, I should evaluate the following…
Marco Gandolfi
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Fourier transform of a triangle

Consider a 2-dim regular n-gon whose vertices lie on the unit circle. Let $\chi_n$ denote the characteristic function of this polygon and $\widehat{\chi}_n$ its Fourier transform. The special case n = 4 lends itself particularly well to…
user2052
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Greens function of Laplace operator

I am having trouble deriving the Greens function for the three dimensional Laplacian $\nabla^2$. I wish to solve the equation $$ -\nabla_r^2 G(r,r') = \delta(r-r') \quad\quad\quad r,r'\in\mathbb{R}^3$$ I know that $\frac{1}{4\pi|r-r'|}$ is the…
alexvas
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Rewriting kernel using Fourier transform

Show (using the Fourier transform) that $$ K(u) = \frac{1}{2} \exp \left(- \frac{|u|}{\sqrt{2}} \right) \sin \left(\frac{|u|}{\sqrt{2}} + \frac{\pi}{4} \right) $$ can also be written $$ K(u) = \int_{-\infty}^\infty \frac{\cos(2\pi t u)}{1+(2\pi…
Lundborg
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$\int_{0}^{\infty} f(x) \sin(x) dx$ from Complex Fourier Transform for $f(x)$ even

I am able to calculate $\int_{0}^{\infty} f(x) \cos(x)\, \mathrm{d}x$ for $f(x)$ being even by taking the real part of Complex Fourier transform (at $\omega = 1$). The two-sided sine transform is $0$, as $f(x)$ is even. Is there a way to calculate…
Srini
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Compute the Fourier transform of a $L^2$ function.

Let $f\in L^2(\mathbb R)$. Is it possible to compute the Fourier transform of an $L^2(\mathbb R)$ function ? I'm asking this question because in a book I'm reading, they take the Fourier transform of a function $f\in L^2(\mathbb R)$ but for me there…
MSE
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Function that isn't the Fourier transform of periodic integrable function

What's an example of a function $\phi : Z\rightarrow C$ such that $\phi(n) \rightarrow 0$ as $|n|\rightarrow\infty$, and $\phi$ is not the Fourier transform of some function $f\in L^1([-\pi,\pi])$? Rudin proves in Real and Complex Analysis that…
user3927913
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Fourier transform in $\mathbb{R}^n$ of $e^{-\|x\|}$

Given $n\in\mathbb{N}$, I'm stuck in the calculation of the Fourier transform in $\mathbb{R}^n$ of $$(x_1,\dots,x_n)\mapsto\exp(-\sqrt{x_1^2+\dots+x_n^2}).$$ I know how to compute the result for each $n$ odd (see e.g. [1]: Fourier transform of…
Bob
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Fourier Transform in spherical coordinates

I have a problem finding the Inverse Fourier transform of the $$e^{i\bf{k}\cdot\bf{r}}\dfrac{1}{k^2+a^2}$$ Any suggestion?
Jon Snow
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Fourier transform of $ \int_{-\infty}^{t} f(\eta )\text{d}\eta $

Suppose $f(t)$ and $F(\omega)$ are a Fourier transform pair. I want to show that $$\mathcal{F}^{-1} \left\{\frac{F(\omega)}{i\omega}\right\} = \int_{-\infty}^t f(\eta)\ \text{d}\eta$$ I start with the Fourier transform of the RHS and use integration…
AtticusFinch95
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Fourier transform of $\exp{\left(-\sqrt{x^2 + 1}\right)}$?

Is there any known "closed form" expressions for the following Fourier transform? $$\int_{-\infty}^{\infty}e^{itx}e^{-\sqrt{x^2 + 1}}dx$$ It is clear that the integral converges: $$\int_{-\infty}^{\infty}e^{itx}e^{-\sqrt{x^2 + 1}}dx =…
martin
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Fourier inversion

For $f\in L^1(\mathbb R^d)$ look at $f_n:\mathbb R^d \rightarrow \mathbb K, x\mapsto\int_\mathbb {R^d}\hat{f}(\xi)e^{2\pi i\langle\xi, x\rangle} e^{-\frac{\pi^2}{n}|\xi|^2}d\xi$. Show $\lim\limits_{n\to\infty} \|f_n-f\|_1=0$ $\hat{f}(x)$ is the…
andy
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