# Questions tagged [fourier-restriction]

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14 questions
0answers
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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 ...
0answers
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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 $\xi\mapsto \xi^2$, $\xi\in [-1,1]$. (...
0answers
47 views

### 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?
1answer
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 ...
2answers
3k views

### 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. ...
0answers
117 views