Question

Find the complex Fourier series of the $2\pi$ periodic function $f(x)=x^2$ and determine the function which the Fourier series converges to.

score 790 · Answer 1 · 2026-03-29 10:48:56

790

Views

Find the complex Fourier series of the $2\pi$ periodic function $f(x)=x^2$ and determine the function which the Fourier series converges to.

Published on 29 Mar 2026 - 10:48

score 119 · Answer 2 · 2026-03-25 14:25:58

119

Views

Region of convergence of the Laplace transform

Published on 25 Mar 2026 - 14:25

score 730 · Answer 3 · 2018-04-03 18:32:30

730

Views

Fourier series for $f(x+\pi) = f(x)$

Published on 03 Apr 2018 - 18:32

score 312 · Answer 4 · 2026-03-25 06:12:59

312

Views

Limit of sequence of finite rank operators given by $(A_Nf)(x) = \sum_{n=-N}^N \hat f(n) e^{inx}$ is the identity operator, so how can it be compact?

Published on 25 Mar 2026 - 6:12

score 37 · Answer 5 · 2026-03-25 22:05:00

37

Views

Prove that $\lim_{N \to \infty} \left( \int_{\pi-\pi/(N+1/2)}^{\pi} \frac{\sin((N+1/2)t)}{\sin(t/2)} dt \right)= 0$

Published on 25 Mar 2026 - 22:05

score 47 · Answer 6 · 2018-04-03 21:48:15

$\{f \in L^1(\mathbb R^n) \cap L^2(\mathbb R^n): \mathcal Ff \in L^2(\mathbb R^n)\}$ dense in $L^1(\mathbb R^n) \cap L^2(\mathbb R^n)$?

score 74 · Answer 7 · 2026-03-25 12:21:14

74

Views

The metric on L2 space expressed in terms of Fourier sine coefficients

Published on 25 Mar 2026 - 12:21

score 491 · Answer 8 · 2026-03-25 19:05:28

491

Views

Find Fourier Series of $f(x)=e^x+e^{-x}$

Published on 25 Mar 2026 - 19:05

score 139 · Answer 9 · 2026-03-27 10:32:42

139

Views

An elementary (interpolation) inequality?

Published on 27 Mar 2026 - 10:32

score 660 · Answer 10 · 2026-03-26 07:32:16

660

Views

On the asymptotic behavior of the Fourier coefficients of $(1-R\cos\theta)^{-3/2}$

Published on 26 Mar 2026 - 7:32

score 1.7k · Answer 11 · 2026-03-25 23:40:30

1.7k

Views

How to prove that the span of $\cos((n+1/2)x)$ and $\sin(nx)$ is dense in $L^2(-\pi,\pi)$?

Published on 25 Mar 2026 - 23:40

score 42 · Answer 12 · 2026-03-26 21:08:58

42

Views

Sovling classical parabolic equation by using Littlewood-Paley theory

Published on 26 Mar 2026 - 21:08

score 53 · Answer 13 · 2026-03-25 23:36:09

53

Views

Evaluating integrals of periodic functions

Published on 25 Mar 2026 - 23:36

score 146 · Answer 14 · 2026-03-22 06:22:28

146

Views

What is the Fourier transform of a "geometric delta comb?"

Published on 22 Mar 2026 - 6:22

score 97 · Answer 15 · 2026-03-26 04:14:13

97

Views

Stationary phase analysis for a Schwartz function

Published on 26 Mar 2026 - 4:14

List Question

Find the complex Fourier series of the $2\pi$ periodic function $f(x)=x^2$ and determine the function which the Fourier series converges to.

Region of convergence of the Laplace transform

Fourier series for $f(x+\pi) = f(x)$

Limit of sequence of finite rank operators given by $(A_Nf)(x) = \sum_{n=-N}^N \hat f(n) e^{inx}$ is the identity operator, so how can it be compact?

Prove that $\lim_{N \to \infty} \left( \int_{\pi-\pi/(N+1/2)}^{\pi} \frac{\sin((N+1/2)t)}{\sin(t/2)} dt \right)= 0$

$\{f \in L^1(\mathbb R^n) \cap L^2(\mathbb R^n): \mathcal Ff \in L^2(\mathbb R^n)\}$ dense in $L^1(\mathbb R^n) \cap L^2(\mathbb R^n)$?

The metric on L2 space expressed in terms of Fourier sine coefficients

Find Fourier Series of $f(x)=e^x+e^{-x}$

An elementary (interpolation) inequality?

On the asymptotic behavior of the Fourier coefficients of $(1-R\cos\theta)^{-3/2}$

How to prove that the span of $\cos((n+1/2)x)$ and $\sin(nx)$ is dense in $L^2(-\pi,\pi)$?

Sovling classical parabolic equation by using Littlewood-Paley theory

Evaluating integrals of periodic functions

What is the Fourier transform of a "geometric delta comb?"

Stationary phase analysis for a Schwartz function

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