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Questions tagged [fourier-restriction]

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Fourier tranform of the derivative

I have been recently studing Fourier transform and there is a proposition that says: If $\lim \limits_{x \to\infty}xf(x)=\lim\limits_{{x}\to -\infty}xf(x)=0$ then $$\hat{f'}(z)=-iz\hat{f}(z)$$ and ...
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Fourier Restriction: extension operator identity

Consider the extension operator: $$ Eg(x)=\int_S g(\xi)e^{2\pi i x\cdot \xi}d\sigma(\xi). $$ For simplicity we consider the 2D-case, where $S$ is the paraboloid $\xi\mapsto \xi^2$, $\xi\in [-1,1]$. (...
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Fourier transforms in Real line

If a function $f \in L^{p}([a,b])$ then is it possible that it's Fourier transform $\widehat{f} \in L^{p}(\mathbb{R})$. How can we show this?
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260 views

Trying to show that columns of Fourier matrix are eigenvectors

I posted a picture since the syntax for this one seems quite complex: I found this: Discrete Fourier Transform - proof that columns of matrix are orthogonal which only shows that are orthogonal. In ...
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2answers
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What it the fourier transform of laplacian and shifted funtion?

I'm looking for the Fourier transform of $\nabla^2f(\vec{r}-\vec{a})$ I can assume that the 3D Fourier transform of $f(\vec{r})$ is $\tilde{f}(\vec{q})$ and the vector $\vec{a}$ is a const vector. ...
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Resolvent of the operator

Consider the Laplace operator defined on the biggest possible subset of$L^{2}(R^{2})$: $T= - \partial^{2}_{x} -\partial^{2}_{y}+x^{2}+y^{2}+ 2.i(x \frac{\partial}{\partial y}-y\frac{\partial}{\...
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1answer
123 views

What if we change one of Fourier's law of heat conduction

I'm studying PDE heat diffusion on 1-D rod using the textbook. It states four intuitions leading to Fourier's law of heat conduction $\phi=-K_0\frac{\partial u}{\partial x}$, where $\phi$ is the heat ...
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105 views

Zero Gaussian curvature and restriction estimates

Let $$ \left(\int_M|\hat{f}|^qd\mu(\xi)\right)^{1/q}\leq c||f||_{L^p(\mathbb{R}^n)} $$ be a restriction estimate for a hypersurface $M\subset\mathbb{R}^n,~1<p<\infty$ and $\mu$ a ...
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43 views

Fourier Series Relation Time - Frequency

I want to study and understand the relation between time and frequency with the help of Fourier Series. Can you indicate me some papers, or some example?
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1answer
423 views

Decay of the Fourier transform of the surface measure of the sphere via uncertainty

I'm working through Tao's Recent Progress on the Restriction Conjecture notes (http://arxiv.org/abs/math/0311181). Currently, I'm working on problem 2.4, which will eventually allow us to compute the ...
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1answer
111 views

Equivalence of Localized Fourier Restriction Estimates

I'm reading Tao's Park City notes on the restriction conjecture http://arxiv.org/abs/math/0311181. He says at some point that the estimate: there is a constant $C > 0$ such that, for any $R \ge 1$...
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1answer
114 views

Fourier Transform of exp(-a|x-.5|)

So I've been working on the fourier transform of $exp(-a|x-\frac{1}{2}|)$ (with $a>0$) and keep getting: $\left(e^{-\pi i}\right)\left(\frac{2a}{a^+4\pi^2x^2}\right)$. A research partner keeps ...
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2answers
280 views

Dirac Comb Times Step Function

Can someone explain me what is the effect of the Heaviside step function $\Theta(t)$ on a Dirac Comb (Fourier series)? $$ \left[\,\,\sum_{n=-\infty}^{\infty}c_{n}\,\delta\left(t - nT_{0}\right) \...
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Fourier-restriction on Hyperplane

In Muscalu, Schlag "Classical and multilinear harmonic analysis, Volume 1" (2013), Excercise 11.1 is to prove, basically, that there exists a function $f\in L^p \quad \forall\ p>1$ such, that the ...