0
votes
0answers
38 views

Proving pointwise convergence of Fourier series [on hold]

I'm trying to prove the pointwise convergence of a $2\pi$-periodic function $f$ to its Fourier expansion. The proof on my lecture notes stops at this formula: $$f(t)-P_{N,f}=\frac 1 {2 \pi} ...
4
votes
2answers
59 views

what is the sum of this series: $\frac{2}{\pi}\Sigma_{k=1}^{\infty}\frac{(-1)^{k-1}}{2k-1}$

Can anyone help me with this? What is the sum of this series: $\frac{2}{\pi}\Sigma_{k=1}^{\infty}\frac{(-1)^{k-1}}{2k-1}$ I got it after plugging $x=-1$ in a fourier series Thank you!
1
vote
0answers
49 views

Fourier series using summation methods

My question is similar to this one. There are ways of deriving the formulae like $$\sum_{k = 1}^\infty \frac{\sin(kz)}{k} = \frac{\pi - z}{2}$$ using summation methods. My question is: How can we ...
0
votes
2answers
31 views

Filter on Fourier Series

i have a lowpass filter H(ω) which is $ H(ω) = e^{-jω} $ on -2π≤ω≤2π, and $0$ elsewhere and i have a function in fourier series y(t), i need to find the new signal (z(t)) after the application of the ...
3
votes
1answer
76 views

Fourier series without Fourier analysis techniques

It is known that one can sometimes derive certain Fourier series without alluding to the methods of Fourier analysis. It is often done using complex analysis. There is a way of deriving the formula ...
3
votes
1answer
49 views

Will Fourier Series converge even if you only use Prime Integer frequencies?

So there is a Fourier Series for a function $f(x)$ with period $P$: $$ f(x) = \frac{A_0}{2} + \sum_{n=1}^{N} A_n \cdot \cos \left(\frac{n 2 \pi x}{P} + \phi_n \right) $$ Let $\frac{2 \pi x}{P} = t$ ...
2
votes
1answer
46 views

Series expansion of $\coth x$ using the Fourier transform

Hi I have research about the series of coth but all of the solutions emerges from integral on a contour, Could you calculate the fourier transform of coth? Is that possible at all?My goal is to reach ...
3
votes
0answers
64 views

How to do this Sum? Poisson Resummation?

In the paper hep-th/0812.2909 page 34-35, there's a sum that I've been trying to do explicitly but I can't find a way. The sum is $$ \frac{2l}{\pi l! (l-1)!} \sum_{k\in\mathbb{Z}} \sum_{n=0}^{\infty} ...
4
votes
1answer
62 views

Question regarding Fourier coefficients

I would like to express the product $$ \left( \sum_{k \in \mathbb{Z}} a_k \sin(k t) \right) \left( \sum_{k \in \mathbb{Z}} b_k \cos(k t) \right) $$ as $$ \sum_{k \in \mathbb{Z}} c_k \sin(k t). $$ ...
2
votes
0answers
22 views

A proof regarding Fourier-Polynoms

I want to prove the following: Let $f:\mathbb{R}\rightarrow \mathbb{C}$ so that $f \big |_{[0,2\pi]}$ is integrable. Let $V$ be the vectorspace of all $2\pi$-periodic functions and $U \subset V$ be ...
2
votes
1answer
54 views

What is this waveform?

Consider the following infinite series: $\text{f} \left( x \right) =\displaystyle \sum \limits_{n=1}^{\infty} \frac {\sin \left( n x\right)}{n^2}$ We know that $\text{f} \left( x \right)$ is ...
4
votes
1answer
105 views

Let $f(x) = |\cos(x)|$. Prove the corresponding Fourier series converge point-wise or uniform and show identity.

Consider $f(x) = |\cos(x)|$ for $x \in \mathbb R$. I've proved the n'th fourier coefficient $c_n = \int^{\pi}_{-\pi} f(y)e^{-iny} \ dy = \frac 1 {2\pi} \frac {(-1)^{n-1}} {n^2-\frac 1 4}$. However, ...
1
vote
0answers
40 views

Fourier Series and sum help

I have to find the Fourier series expansion of the function $f(x)$=$x^2$ for $-\pi <x< \pi$ and using it I have to show that, i) $1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}...$ = ...
1
vote
2answers
23 views

How to see this step in deriving an equality in Fourier series?

(Previous steps are omitted.) By convergence of Fourier series, we have $$ \frac{\pi}{4}+\sum_{n=1}^{\infty}\frac{(-1)^n-1}{\pi n^2}(-1)^n=\frac{\pi}{2} $$ Then how come we can get this from the ...
1
vote
0answers
64 views

Fourier Series; odd and even half-range expansion

I have some standard Fourier series questions which I cannot solve. My fourier series is defined like this: $$s(x)=\frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos (nx) + b_n \sin (nx))$$ For $f(t) = ...
1
vote
1answer
110 views

How to find the Fourier series of a periodic function

Find the Fourier series of the function $f(t)=3t^2$, $-1\le t\le 1$. How do I solve this problem? What is the general formula and the way to solve this?
2
votes
0answers
43 views

Show sum involving sines is non-negative

I want to show that \begin{equation} \sum_{\substack{k \geq 1 \\ k \text{ odd}}} k e^{-k^2 a} \sin(kx) \geq 0 \qquad \text{for all } x \in [0,\pi], \, a > 0. \end{equation} How should I start? I ...
0
votes
1answer
24 views

Expanding unity in terms of orthogonal functions cos( alpha(i) * y)

It is written in the book I am reading without proof that if we expand unity in terms of orthogonal functions cos( alpha(i) * y), we get: (Please check this link) ...
0
votes
0answers
13 views

Nyquist Frequency, filter, phase/amplitude

Problem The problem seems quite simple and I believe it is though I have no idea how to approach it. I have tried googling 'Nyquist frequency' but have not had any luck with problems similar to this. ...
2
votes
2answers
118 views

Summing a series (from a physics problem)

How might we show that $$\sum_{k = 0}^{\infty}\frac{2}{2k + 1}e^{-(2k + 1)\pi x/a}\sin\left( \frac{(2k + 1)\pi y}{a}\right) = \tan^{-1}\left( \frac{\sin(\pi y/a)}{\sinh(\pi x/a)} \right) $$ where $x, ...
0
votes
1answer
51 views

Infinite trigonometric series, find the constant C_n

Hi this is my first post :) I am not sure how to do part b. You get the infinite series of $\displaystyle c_n\cdot \sin(\frac{n\cdot \pi\cdot x}{L})$ from $n=1$ to infinity And this is equal to ...
1
vote
0answers
47 views

How to find the value of this sum?

The sum below numerically (to 13th digit at least) is the same as $\ln 2$. So there should be a way to prove it analytically, but I haven't succeeded. Any suggestions? ...
-1
votes
2answers
45 views

Expanding a piecewise defined function, what will the series converge to at $x=-1,0,1$? [closed]

If we expand $$f(x)=\begin{cases} (x+1) & -1<x<0; \\ -x & 0<x<1 \end{cases}$$ what will the series converge to at $x=-1$, $x=0$, and at $x=1$? Hey I tried to work this out on ...
3
votes
1answer
83 views

Fourier series to calculate infinite series

I try to show that $\sum_{i=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$ using Fourier series and $f(x) = x$ on $L^2_{\mathbb{C}}[-\pi, \pi]$, with basis $e_n(x) = \frac{1}{\sqrt{2\pi}}e^{inx}$. I ...
11
votes
2answers
858 views

Why is this allowed? (“Fourier's Trick”; finding the coefficients in a Fourier Series)

In my textbook (Introduction to Electrodynamics, D. Griffiths), we derive the equation for some strange potential function. Eventually, we get to this (for $n \in \mathbb{Z}^+$): $$ V_0(y) = ...
3
votes
4answers
144 views

Find the Exact sum

Give the fourier series representation of $f(x) = x$ on $[-\pi, \pi]$. Use the result to give the exact sum of... $$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{2n-1}$$ $$\text{ where } x \in [-\pi,\pi]$$
0
votes
0answers
27 views

Discrete Fourier Transform of the infinite series

I am reading this book and having hard time understanding how to get to eq(2) from eq(1) $$P(k,t) = e^{-\alpha t} \sum\limits_{l,m=-\infty}^\infty (-i)^m e^{ik(l+m)} I_l(\alpha t) I_m(i\beta t) ...
1
vote
2answers
54 views

Find the fourier series for $\cos^{2N}(\theta )$.

I'm working my way through a book for prelim prep and found the problem: Find the fourier series for $\cos^{2N}(\theta )$. The hint is to not use integrals but the only method I know involves ...
7
votes
2answers
232 views

Compute $\sum\limits_{n = 1}^{\infty} \frac{1}{n^4}$.

Compute the Fourier series for $x^3$ and use it to compute the value of $\sum\limits_{n = 1}^{\infty} \frac{1}{n^4}$. I determined the coefficients of the Fourier series, which are $$a_0 = ...
3
votes
1answer
112 views

Convergence of series of functions: $f_n(x)=u_n\sin(nx)$

Let $f_n(x)=u_n\sin(nx)$ where $\displaystyle\sum f_n$ converges pointwise, and $ \displaystyle x \mapsto \sum_{n=0}^{+\infty} f_n(x)$ is continuous. Prove that $ u_n\rightarrow 0$ when n ...
5
votes
2answers
812 views

Evaluate $\sum^\infty_{n=1} \frac{1}{n^4} $using Parseval's theorem (Fourier series)

Evaluate $\sum^\infty_{n=1} \frac{1}{n^4} $using Parseval's theorem (Fourier series). I have , somehow, to find the sum of $\sum_{n=1}^\infty \frac{1}{n^4}$ using Parseval's theorem. I tried ...
2
votes
1answer
248 views

Fourier Series of $f(x) = 0$ from $(-\pi, 0)$, $x$ from $(0,\pi)$

I need to determine the fourier series of the following function, (using trig method, not complex) $$ f(x) = \begin{cases} 0 & \text{if } -\pi < x < 0, \\ x & \text{if } 0 < x < ...
4
votes
3answers
176 views

Evaluate $ \sum_{n=1}^{\infty} \frac{\sin \ n}{ n } $ using the fourier series

I am a beginer with Fourier series and i have to evaluate the sum $$\sum_{n =1}^{\infty}{\sin\left(n\right) \over n}$$ I dont know which function i have to take to evaluate the fourier series ... ...
2
votes
2answers
93 views

Expansion and convergence of $\sum_{m=1}^{\infty}\frac{\sin(2\pi n x^{1/m})}{ m}$

Consider the series: $$\sum_{m=1}^{\infty}\frac{\sin(2\pi n x^{1/m})}{ m}\;\;\;\;n\in\mathbb{N}$$ Other than formal manipulation of the Taylor series of the $\sin$ function, is there a way to expand ...
1
vote
1answer
56 views

Fourier Series Approximations of Functions

From a few examples of smooth functions, discontinuous functions and continuous functions which have a 'kink' (i.e. $|x|$ where left and right limits disagree)... I've seen that the fourier series ...
2
votes
1answer
112 views

Calculate the sum using Fourier series $\big|\cos(\frac{x}{2})\big|$.

I have been given task to evaluate the sum $\frac{(-1)^{n-1}}{4n^2-1}$ using Fourier series for function $|\cos{\frac{x}{2}}|$. I used the interval $(-\pi, +\pi)$ for evaluation of the sum and I ...
3
votes
1answer
138 views

Parseval's Identity (Integral)

Calculate the integral: \begin{equation} \int_{-\pi}^{\pi}\left|\sum_{n=1}^{\infty}\frac{1}{2^{n}}e^{inx}\right|^{2}dx\end{equation} I'm familiar with Parseval's identity which states that for ...
3
votes
1answer
53 views

Show that for $0<t<1$, $\log\sin (t\pi)=\log(t\pi) + \sum_{n=1}^\infty\log\left(1-\frac{t^2}{n^2}\right)$

Show that for $0<t<1$, $$\log\sin (t\pi)=\log(t\pi) + \sum_{n=1}^\infty\log\left(1-\frac{t^2}{n^2}\right)$$ So I derived the following Fourier series: ...
2
votes
0answers
25 views

Convergence of the series $\sum_{\xi\in\mathbb Z^n} e^{2\pi ix\cdot \xi} a(x, \xi)\hat{f}(\xi)$?

I need some help with the following problem: let $a:\mathbb R^n\times \mathbb R^n\rightarrow \mathbb C$ be a smooth function and suppose there are constantes $C_{\alpha, \beta}$ and $M(\alpha, \beta)$ ...
1
vote
1answer
67 views

Show absolute and uniform convergence of a Fourier series

Hello and good evening! The Fourier series of $f(x):=\lvert x\rvert$ on $[-\pi,\pi]$ is $$ f(x)=\frac{\pi}{2}-\frac{4}{\pi}\sum_{n=1}^{\infty}\frac{\cos((2n-1)x)}{(2n-1)^2}. $$ I have to examine if ...
3
votes
1answer
202 views

Basic Fourier Series Question

Let $f$ be a $2π$ periodic function where $$f(x) = \frac{π - x}2$$ over $[0, π]$. It is known that the Fourier series of $f$ is $$\sum_{n=1}^{\infty}\frac{\sin nx}n$$ At which points in $[-π, π]$ ...
9
votes
1answer
227 views

Is $\sum_{n=1}^{\infty}{\frac{\sin(nx)}n}$ continuous?

Considering the infinite series $\sum_{n=1}^{\infty}{\frac{\sin(nx)}n}$ , I can show that it is not convergent uniformly by Cauchy's criterion and that it is convergent for every $x$ by Dirichlet's ...
2
votes
2answers
51 views

$C$ such that $\sum_{k\in \mathbb{Z}^n} k_i^2k_j^2|a_{ij}|^2 \leq C \sum_{k\in \mathbb{Z}^n} \|k\|^4|a_{ij}|^2$

More generally, can we find $C_n>0$ such that $$\sum_{k\in \mathbb{Z}^n} k_i^2k_j^2|a_{ij}|^2 \leq C \sum_{k\in \mathbb{Z}^n} \|k\|^2|a_{ij}|^4$$ for all $\{a_k\}_{k\in \mathbb{Z}^n} \in ...
2
votes
1answer
106 views

Prove that the Fourier series of $\dfrac{1}{f}$ is absolutely convergent.

I have a problem: Let $f$ be a continuous function on the unit circle $(\Gamma)$: $$\Gamma=\{e^{i\theta}: \theta\in [0, 2 \pi]\}$$ Assume that $f \ne 0$ on $\Gamma$, and the Fourier ...
0
votes
1answer
68 views

Prove that $\sum\limits _{n=-\infty}^{n=\infty}\cos\left(2\pi nt\right)=\sum\limits _{n=-\infty}^{n=\infty}\delta\left(t-n\right)$

I've tried using Fourier transforms on both but didn't quite get anything useful. I'd really appreciate some help.
1
vote
1answer
644 views

fourier series of $|\sin x|$

I need to find the fourier series of $$|\sin x|$$. Im not sure my way is right, would be happy if someone fix me. I found $$a_0=4/\pi$$, the function is even, so $$b_n=0$$ but how do I calculate: ...
0
votes
1answer
32 views

Does this transformation have an inverse?

Let $f(n)$ be a complex sequence. Then for prime $p$ define $\hat{f}(p) = \sum_{n = 1}^{\infty} a_n e^{-i 2 \pi n / p}$. Then let the transformation of sequences be $T$, i.e. $Tf = \hat{f}$. Is ...
0
votes
0answers
32 views

Working with infinite sequences in $\ell^2(\mathbb{Z})$

Let $\ell^2(\mathbb{Z})$ be the set of all two-sided sequences $(a_i)$ in $\mathbb{C}$, such that $\sum_{n\in \mathbb{Z}} |a_n|^2 \lt \infty$. What considerations do I have to take into account when ...
0
votes
1answer
77 views

The Fejer kernel has this $\sin$ closed form.

Let $D_N$ be the $N$th Dirichlet kernel, $D_N = \sum_{k = -N}^N w^k$, where $w = e^{ix}$. Define the Fejer kernel to be $F_N = \frac{1}{N}\sum_{k = 0}^{N-1}D_k$. Then $$F_N = ...
4
votes
5answers
289 views

If $f_n \rightarrow f$ and $f$ is uniformly continuous, then does $f_n \rightarrow f$ uniformly?

Let $f_n$ be a sequence of functions, $f_n : S\rightarrow T$, not necessarily continuous and suppose that $f_n \rightarrow f$ as $n \rightarrow \infty$. Let $f$ be uniformly continuous. I.e. for ...