All Questions
14
questions
1
vote
0
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38
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When I was looking at the proof of Fejer's theorem, I encountered a problem in the derivation of a formula. [closed]
How did we get the last equation? Why can the summation be converted into a square term?
$$
\begin{align}K_m(x)&:=1+\frac{2}{m}\sum_{j=1}^{m-1}(m-j)\cos(jx)
\\&= \frac1m\sum_{j=-(m-1)}^{m-1} (...
- 21
1
vote
1
answer
184
views
Finding the sum of a series using a Fourier series
I am stuck on how to calculate the value of the following sum:
$\sum_{n=0}^\infty \frac{(-1)^n}{2n+1}$
I am aware that you need to find the corresponding function whose Fourier series is represented ...
- 351
1
vote
0
answers
20
views
proofs about a norm involving an integral and fourier coefficients
Let $C([-\pi, \pi])'$ be the set of continuous functions $f$ from $[-\pi, \pi]$ to $\mathbb{C}$ such that $f(\pi) = f(-\pi)$. For $f \in C([-\pi, \pi])', n \in \mathbb{Z}$, define $a_n = \frac{1}{2\pi}...
- 1,225
3
votes
1
answer
194
views
Tauber's theorem (Abel summable $\implies$ convergent) for $\sum c_n$ where $\lim_{n\to\infty} nc_n = 0$
Prove that if $\sum_n c_n$ is Abel summable to $s$, and $nc_n\xrightarrow{n\to\infty} 0$, then $\sum_n c_n$ converges to $s$.
Exercise $14(b)$, Chapter $2$, Stein & Shakarchi's Fourier Analysis.
...
- 14.2k
0
votes
1
answer
231
views
Prove that $\sum_{n=-\infty }^{+\infty }\frac{1}{(n+\alpha )^{2}}=\frac{\pi ^{2}}{(\sin\pi \alpha )^{2}}$ with poisson summation formula [duplicate]
I want to show that $\sum_{n=-\infty }^{+\infty }\frac{1}{(n+\alpha )^{2}}=\frac{\pi ^{2}}{(\sin\pi \alpha )^{2}}$, the introduction to the Poisson summation formula is in this link https://en....
- 623
3
votes
2
answers
351
views
Whats is the sum of series for each $0\le x\le 3$?
Let $\psi(x)=\begin{cases}0:& 0<x<1\\ 1:& 1<x<3 \end{cases}$
a) Compute the first 4 terms of its Fourier cosine series explicitly.
b) For each $x (0\le x\le 3)$, whats is the sum ...
- 1,658
3
votes
1
answer
581
views
Taylor expansion and Distribution
Let $u(x)$ be the step function and $p_u(x)$ be the distribution defined by
$$\forall \varphi \in D,
\langle p_u , \varphi \rangle
= \lim_{\epsilon \to 0} \left (\varphi(0) \ln(\epsilon) + \int^{...
- 1,940
4
votes
1
answer
126
views
Is it possible to interchange the order of limits in this case?
I'm stuck on this problem.
Let $g \in C^{\infty}(\mathbb{R})$ with $|x|^p |D^{q} g| \rightarrow 0$ as $|x| \rightarrow \infty$ for any nonnegative integers $p$ and $q$.
Suppose that $|g(\gamma)| \...
- 660
3
votes
0
answers
411
views
Cauchy Product of Fourier Series with itself
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be an $L^1$ and $L^2$-integrable function over $[0,2\pi)$ whose Fourier coefficients are real and are given by $\hat{f}$, and in particular $\hat{f}(0) = 0$. ...
- 1,145
4
votes
0
answers
427
views
Convergence of sum involving Bessel function of the first kind
I am interested in finding any upper bound for the following series:
$$\displaystyle f(x) := \sum_{n \in \mathbb{Z}^d \backslash \{0\}} |n|^{-d/2}J_{d/2}(k|n|)e^{inx},$$
where $d \geqslant 2$, $x \...
- 1,145
0
votes
1
answer
245
views
Recurrence equation from an infinite Fourier series
Consider the following set of equations
$$
\sum_{k=-\infty}^{\infty} \left[ \psi_k(\lambda) \left( \lambda I_k'(\lambda R) \cos k\phi \cos \phi + \frac{k}{R} I_k(\lambda R) \sin k\phi \sin \phi \...
3
votes
1
answer
1k
views
Identity for the sum of products of Sinc functions
The Sinc function is defined as follows:
$$\mathrm{sinc}(t) = \begin{cases} \frac{\sin(\pi t)}{ \pi t} & \mathrm{if} \quad t \neq 0, \\ 1 & \mathrm{otherwise.} \end{cases}$$
I want to show the ...
- 321
0
votes
1
answer
40
views
Simplification of a large sum obtained from the 1-D wave equation
I have acquired the sum below through Fourier, and was wondering if there was anyway to simplify it, since it is large and ugly.
$$\sum \limits_{n=1}^\infty \frac{-2K_1}{n\pi} \left(\cos\left(\frac{...
user142198
4
votes
1
answer
77
views
How to tell which function's Fourier series to use in order to calculate the value of series.
I got this question when I was doing some exercises. I was ask to establish $$
\sum_{n=0}^{\infty}\frac {1}{(2n+1)^2}=\frac{\pi^4}{96},\quad \sum_{n=0}^{\infty}\frac {1}{(2n+1)^6}=\frac{\pi^6}{960},\...
- 13.3k