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The Fourier transform of $ f(x)=xe^{{-x}^2} $

score 16.3k · Answer 1 · 2026-04-14 21:52:36

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The Fourier transform of $ f(x)=xe^{{-x}^2} $

Published on 14 Apr 2026 - 21:52

score 491 · Answer 2 · 2026-04-13 04:23:16

491

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fourier coefficients and characteristic

Published on 13 Apr 2026 - 4:23

score 616 · Answer 3 · 2026-04-12 13:20:54

616

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Applications of Euclidean harmonic analysis to geometry?

Published on 12 Apr 2026 - 13:20

score 1.5k · Answer 4 · 2026-04-15 06:30:30

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Closed form of the Dirichlet kernel: $\sum\limits_{n=-N}^{N} e^{inx} =\frac{\sin((N+1/2)x}{\sin(x/2)}$

Published on 15 Apr 2026 - 6:30

score 1000 · Answer 5 · 2026-04-16 20:37:56

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Lower $ L^{1} $ bound for the Dirichlet kernel: $ \frac{1}{2\pi}\int _0^{2\pi} \left| \frac{\sin((N+1/2)x)}{\sin(x/2)}\right| dx \geq C \log N $

Published on 16 Apr 2026 - 20:37

score 164 · Answer 6 · 2026-04-14 04:48:12

164

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Variants of Parseval's relation

Published on 14 Apr 2026 - 4:48

score 54 · Answer 7 · 2026-04-11 20:03:58

54

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How to show that $e^{-ax^2}$ is an element of $\mathcal{S}(\mathbb{R})$?

Published on 11 Apr 2026 - 20:03

score 144 · Answer 8 · 2026-04-17 03:18:37

144

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If for every $t \in \mathbb{R}$, $\left\vert \int_{\mathbb{R}}f(x) e^{-2 \pi ix \cdot t} dx \right\vert < \infty$, is $f(x) \in L^{1}(\mathbb{R})$?

Published on 17 Apr 2026 - 3:18

score 613 · Answer 9 · 2026-04-11 22:31:01

613

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Question from Stein's Fourier Analysis on sequence of integrable functions

Published on 11 Apr 2026 - 22:31

score 78 · Answer 10 · 2026-04-12 19:48:47

78

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Majorizing Sharp Truncation of Integral by a Smooth Truncation

Published on 12 Apr 2026 - 19:48

score 2.5k · Answer 11 · 2026-04-12 05:34:41

2.5k

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Why does convolution become multiplication when taking the Fourier transform?

Published on 12 Apr 2026 - 5:34

score 38 · Answer 12 · 2026-04-15 21:28:45

$\lim\limits_{\delta_j\to0}\int\limits_{\delta_j\le x\le 1}K(x)$ exists $\iff \lim\limits_{\delta_j\to0}\int\limits_{\delta_j\le x}K(x)\phi(x)$

score 198 · Answer 13 · 2026-04-12 01:43:43

198

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How to show that the gaussians are an approximation of the identity?

Published on 12 Apr 2026 - 1:43

score 39 · Answer 14 · 2026-04-13 21:29:05

39

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Intuition about the definition of "belongs to $\mathcal{S}(\mathbb{R})$ uniformly in $t$"

Published on 13 Apr 2026 - 21:29

score 225 · Answer 15 · 2026-04-12 22:55:16

225

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How do I show that the integral of $e^{inx}$ over a set of measure $1$ is nonzero for some nonzero $n$?

Published on 12 Apr 2026 - 22:55

List Question

The Fourier transform of $ f(x)=xe^{{-x}^2} $

fourier coefficients and characteristic

Applications of Euclidean harmonic analysis to geometry?

Closed form of the Dirichlet kernel: $\sum\limits_{n=-N}^{N} e^{inx} =\frac{\sin((N+1/2)x}{\sin(x/2)}$

Lower $ L^{1} $ bound for the Dirichlet kernel: $ \frac{1}{2\pi}\int _0^{2\pi} \left| \frac{\sin((N+1/2)x)}{\sin(x/2)}\right| dx \geq C \log N $

Variants of Parseval's relation

How to show that $e^{-ax^2}$ is an element of $\mathcal{S}(\mathbb{R})$?

If for every $t \in \mathbb{R}$, $\left\vert \int_{\mathbb{R}}f(x) e^{-2 \pi ix \cdot t} dx \right\vert < \infty$, is $f(x) \in L^{1}(\mathbb{R})$?

Question from Stein's Fourier Analysis on sequence of integrable functions

Majorizing Sharp Truncation of Integral by a Smooth Truncation

Why does convolution become multiplication when taking the Fourier transform?

$\lim\limits_{\delta_j\to0}\int\limits_{\delta_j\le x\le 1}K(x)$ exists $\iff \lim\limits_{\delta_j\to0}\int\limits_{\delta_j\le x}K(x)\phi(x)$

How to show that the gaussians are an approximation of the identity?

Intuition about the definition of "belongs to $\mathcal{S}(\mathbb{R})$ uniformly in $t$"

How do I show that the integral of $e^{inx}$ over a set of measure $1$ is nonzero for some nonzero $n$?

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