1
vote
0answers
11 views

Combining even and odd parts of a Chebyshev series

I imagine this will be an easy problem, perhaps even routine, for some. I am learning to manipulate sums and need insight. I started with a power series $$s(x) = \sum_{n=0}^{\infty} a_n x^n$$ and ...
1
vote
1answer
37 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
40 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
18 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
12 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
115 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
47 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 ...
0
votes
0answers
42 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
38 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
74 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
681 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
131 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
23 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
36 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
185 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
83 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 ...
4
votes
2answers
267 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
148 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
143 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
76 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
54 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
93 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
126 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
48 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
61 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
161 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
175 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
98 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
65 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
519 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
62 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
288 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 ...
0
votes
2answers
130 views

The Fourier series converges absolutely $\implies$ it converges uniformly.

Let $S_N(f)$ be the $N$th partial sum of the Fourier series for $f$. I.e. $$S_N(f) = \sum_{n = -N}^{N} \hat{f}(n) e^{2\pi i n x / L}$$ Suppose that the Fourier series converges absolutely, i.e. ...
0
votes
2answers
318 views

Step function Fourier series

How to do a step function based Fourier series? What I am confused about is how to calculate the time period since the step function doesn't end? And I don't really know the period since the ...
2
votes
1answer
40 views

Convergence of eigenmodes of a Sturm Liouville operator.

Is there any "eassy to see" proof for: "The eigenmodes of a Sturm Liouville ODE in a closed interval [a,b], with given boundary conditions, form a complete, orthogonal basis for continuous functions ...
0
votes
0answers
174 views

Find the First Three Nonzero fourier approximations to the following Square Function:

$f(x)= -1$ from $x=-\pi$ to $x=0$ and it is equal to $1$ from $0$ to $\pi$. I know this function is an odd function so only the sine parts of the fourier approximation remain. The first three ...
0
votes
1answer
52 views

Can we estimate the lower bound in this way?

This post is aimed to find a lower bound of $\sum_{k=1}^{n}\frac{\cos(kx)}{k}$ for arbitrary $n \geq 1$ ================================= My approach: Let $S_n(x)$ denote the partial sum of the ...
0
votes
1answer
111 views

Find Fourier Series coefficients of x=1 line function.

I want to know that can we find the Fourier series coefficients of the periodic signal x=1 where the limits are from -infinity to +infinity. The problem arises with the limits and it will converge to ...
1
vote
0answers
30 views

Approximation the function $f(t)=I_0(-rt)e^{-rt}$ with sum of Exponentials.

Consider the function $f(t)=I_0(-rt)e^{-rt}$ where $I_0(t)$ is modified Bessel’s function and $r>0$. I am looking for an approximation for the function with a sum of exponential functions in $t ...
1
vote
0answers
367 views

Represent non-periodic functions in a Fourier Series like function

I have this question of whether it is possible to represent non-periodic functions in a form just like you would represent a periodic function through a Fourier series. I understand this question ...
2
votes
0answers
44 views

Representing series $f(t)= \frac{\pi c^2}{l^2} \sum_{n=1}^\infty \frac{ n }{\omega_n}\cos(\omega_nt)$ as a Dirac comb function.

Consider the function $$f(t)= \frac{\pi c^2}{l^2} \sum_{n=1}^\infty \frac{ n }{\omega_n}\cos(\omega_nt)$$ where $\omega_n= \sqrt{(\frac{n \pi c}{l})^2-(\frac{r_0}{2})^2}.$ If we neglect the term ...
0
votes
0answers
66 views

A general theorem for periodic sequences?

I wonder whether from a very formal point of view one could regard with certainty the following statement on periodic sequences (1) as directly proven through Fourier's theorem (ref. inverse discrete ...
1
vote
0answers
123 views

Fourier series for piecewise function

let $-\pi \leq x\leq \pi$ and $$f(x)=\begin{cases}-x-\pi, & \text{ if} -\pi \leq x\leq -\pi/ 2\\ \;\;\;x, & \text{ if } -\pi/2 \leq x\leq \pi/2\\ -x+\pi, & \text{ if } \pi/2 \leq x\leq ...
1
vote
2answers
1k views

Fourier series for $f(x)=(\pi -x)/2$

I need to find the Fourier series for $$f(x)=\frac{\pi -x}{2}, 0<x<2\pi$$ Since the interval isn't symmetric over $0$, I guess I need to consider $f$'s periodic extension to $\mathbb R$. let's ...
3
votes
1answer
121 views

Fourier series $\sum_{m=0}^\infty \frac{\cos (2m+1)x}{2m+1}$

Does anyone know the sum of Fourier series $$\sum_{m=0}^\infty \frac{\cos (2m+1)x}{2m+1}?$$ I tried WA; it does not return a function.
66
votes
4answers
2k views

Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$

I would like to prove that $\displaystyle\sum_{\substack{n=1\\n\text{ odd}}}^{\infty}\frac{n}{e^{n\pi}+1}=\frac1{24}$. I found a solution by myself 10 hours after I posted it, here it is: ...