Question

Fourier Transform of Powers of Functions

score 663 · Answer 1 · 2026-04-09 08:06:49

663

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Fourier Transform of Powers of Functions

Published on 09 Apr 2026 - 8:06

score 198 · Answer 2 · 2026-04-10 04:22:06

198

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Transition between fourier transforms of periodic and aperiodic signals

Published on 10 Apr 2026 - 4:22

score 96 · Answer 3 · 2026-04-12 17:21:49

96

Views

Solving a wave equation (PDE)

Published on 12 Apr 2026 - 17:21

score 548 · Answer 4 · 2026-04-13 13:51:08

548

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Showing that the equation $f(x)=1$ satisfy the condition of the fourier integral

Published on 13 Apr 2026 - 13:51

score 127 · Answer 5 · 2026-04-16 12:34:31

127

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Delayed means of $f_\alpha (x)=f(x)=\sum_{n=0}^{\infty} 2^{-n\alpha}e^{i2^n x}$ equals its Cesaro mean.

Published on 16 Apr 2026 - 12:34

score 50 · Answer 6 · 2026-04-16 19:12:11

50

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$\sigma_N (g)'(x_0)=-\int_\pi^\pi F_N'(x_0-t)g(t)dt=\int_{-\pi}^{\pi}F_N'(t)g(x_0-t)dt,$ where $F_N$ is the Fejér kernel.

Published on 16 Apr 2026 - 19:12

score 30 · Answer 7 · 2026-04-16 19:09:06

30

Views

Show that if $g$ is differentiabl at $x_0$ we get $|\int_{-\pi}^{\pi}F_N'(t)[g(x_0-t)-g(x_0)]dt|\le C\int_{-\pi}^{\pi}|F_N'(t)||t|dt$.

Published on 16 Apr 2026 - 19:09

score 118 · Answer 8 · 2026-04-15 17:41:52

118

Views

Show that the exponentials $1, e^{2\pi ix}, \dots , e^{2\pi ikx}, \dots$ form the basis for trigonometric polynomials.

Published on 15 Apr 2026 - 17:41

score 987 · Answer 9 · 2026-04-13 12:12:03

987

Views

Prove the inverse Fourier transform of Gaussian kernel is Gaussian distribution (by Bochner's)

Published on 13 Apr 2026 - 12:12

score 775 · Answer 10 · 2026-04-14 14:31:26

775

Views

Proof that the Fourier transform is differentiable for Schwartz space functions

Published on 14 Apr 2026 - 14:31

score 1.1k · Answer 11 · 2026-04-09 21:26:46

1.1k

Views

On "uniqueness" of Fourier expansion

Published on 09 Apr 2026 - 21:26

score 43 · Answer 12 · 2026-04-13 16:30:05

43

Views

How to argue that $f$ is real without assuming $f$ is continuous?

Published on 13 Apr 2026 - 16:30

score 739 · Answer 13 · 2026-04-11 11:27:02

739

Views

Understanding a step in the proof of Fourier Inversion Theorem by Stein

Published on 11 Apr 2026 - 11:27

score 46 · Answer 14 · 2026-04-14 18:35:53

46

Views

Energy of a signal $s(t)=\int_{-4}^{3} e^{{\pi}{i}{t}{u}} du$

Published on 14 Apr 2026 - 18:35

score 1.2k · Answer 15 · 2026-04-12 04:00:40

1.2k

Views

Understanding a step in Stein's proof that the Schwartz space is closed under convolution.

Published on 12 Apr 2026 - 4:00

List Question

Fourier Transform of Powers of Functions

Transition between fourier transforms of periodic and aperiodic signals

Solving a wave equation (PDE)

Showing that the equation $f(x)=1$ satisfy the condition of the fourier integral

Delayed means of $f_\alpha (x)=f(x)=\sum_{n=0}^{\infty} 2^{-n\alpha}e^{i2^n x}$ equals its Cesaro mean.

$\sigma_N (g)'(x_0)=-\int_\pi^\pi F_N'(x_0-t)g(t)dt=\int_{-\pi}^{\pi}F_N'(t)g(x_0-t)dt,$ where $F_N$ is the Fejér kernel.

Show that if $g$ is differentiabl at $x_0$ we get $|\int_{-\pi}^{\pi}F_N'(t)[g(x_0-t)-g(x_0)]dt|\le C\int_{-\pi}^{\pi}|F_N'(t)||t|dt$.

Show that the exponentials $1, e^{2\pi ix}, \dots , e^{2\pi ikx}, \dots$ form the basis for trigonometric polynomials.

Prove the inverse Fourier transform of Gaussian kernel is Gaussian distribution (by Bochner's)

Proof that the Fourier transform is differentiable for Schwartz space functions

On "uniqueness" of Fourier expansion

How to argue that $f$ is real without assuming $f$ is continuous?

Understanding a step in the proof of Fourier Inversion Theorem by Stein

Energy of a signal $s(t)=\int_{-4}^{3} e^{{\pi}{i}{t}{u}} du$

Understanding a step in Stein's proof that the Schwartz space is closed under convolution.

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