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!
0
votes
0answers
9 views

Fourier expansion of sum of an arbitrary function and a trig function

I have this BVP with initial condition being $v(x,0) = -x/\pi - (1/25)sin5x$ and I'm looking for $v(x,t) = \sum b_n sin(nx)e^{-n^2t}$ Expanding $v(x,0)$ gives $v(x,0) = -(1/25)sin5x - ...
1
vote
1answer
55 views

Fourier series problems

I've got an "interesting" problem. I've gotten a way through it, but I'd like someone to look if what I've done so far is correct, and what to do next. We've got a function that is $0$ on the ...
2
votes
0answers
101 views

Prove that periodic analytic function can be written as $\sum_{-\infty}^{\infty} c_n e^{2\pi inz}$

This question involves the following homework problem: PROBLEM Suppose $f$ is analytic in the upper half plane and periodic of period 1. Show that $f$ has an extension of the form ...
4
votes
1answer
69 views

how to compute this integral for fourier series

I am trying to find the Fourier sine and cosine series of $\frac{1}{(1+x^2)}$ from $0$ to $2$, and do not know where to even begin to evaluate this integral: $\int \frac{sin(nx)}{(1+x^2)} dx$ (and ...
2
votes
1answer
78 views

Fourier series of $\sin x$ using series of $e^{ix}$

I have to find the Fourier series of $\sin x$ . Assume that $\ell$ is not an integer multiple of $\pi$.(Hint: First find the Fourier series for $e^{ix}$) This is how I did it: Complex Fourier series ...
1
vote
0answers
80 views

Fourier Series of $f(x)=e^x$ on $[0,\pi)$ as a function of period $\pi$

Can you tell me what you get? I've tried computing it, I've got some result but I don't think it's right since I need to use it for something else and it doesn't work at all... What exactly I'm trying ...
1
vote
3answers
81 views

Calculating own dft via matlab?

We are asked to code our own dft function from the formula : If everything is done correctly it should give the same result with matlab's own dft function, in the end I'm comparing them but they ...
1
vote
1answer
43 views

Find Fourier series coefficients of $f(x)$.

$T=2$ $$f(x) = \begin{cases} 1, & \text{$-\frac12\le x \le\frac12$} \\[2ex] |2x|, & \text{$\frac12 < x \le1\frac12$} \\ \end{cases}$$ The image: I found that $a_0=\frac12$. Since ...
0
votes
0answers
14 views

Pointwise convergence of periodic functions

Let ${f_n}$ be a sequence of functions on $\mathbb{R}$ which satisfy $f_n(x+2 \pi) = f_n(x)$ for all $n$ and $x$. Suppose that $f_n \rightarrow f$. Prove that $f(x+2 \pi) = f(x)$ for all $x$. My ...
0
votes
2answers
25 views

Fourier Transform of $f(t+a)$ if $f(t)$ has tranform $F(k)$?

I know the formula $$f(t) = \int^{+\infty}_{-\infty} F(k)e^{ikt} \, dk$$ and I've seen that for computing $f'(t)$ it's a case of differentiating $e^{ikt}$ inside the integral, so $f'(t)=ikF(k)$ Can ...
0
votes
1answer
33 views

Series expansion Fourier-Legendre

Can anyone explain me how can I expand this function using the Fourier-Legendre expansion? f(x) = x ; -1<=x<=1 Thanks.
0
votes
0answers
20 views

linear algebra - fourier coefficients of piecewise

Find fourier coefficients of given function: f(t) = {-1 if t $\leq$ 0; 1 if t > 0} so do I do this? $a_{0} = \int_{a+-\pi}^{a+\pi}1$, $a_{k} = \int_{a+-\pi}^{a+\pi}1*cos(kx)$, $b_{k} = ...
1
vote
2answers
46 views

fourier series analysis, show that for every integer n, using euler's formulas relating trigonometric and exponential functions

Show that for every integer $n$, $$\int_0^{\pi} \cos nt~\sin t~\mathrm{d}t = \begin{cases} \dfrac{2}{1-n^2} & \text{if } n \text{ is even} \\[10pt] 0 &\text{if } n \text{ is odd} ...
3
votes
1answer
67 views

Poisson summation formula clarification regarding Fejer kernel

Define $$\mathbf{F}_R(t) = \begin{cases} R \left(\dfrac{\sin(\pi R t)}{\pi R t}\right)^2 & t \neq 0\\[10pt] R & t = 0 \end{cases} $$ A problem in Stein's Fourier Analysis asks ...
1
vote
2answers
50 views

Finding complex Fourier coefficients

This is probably an easy question, but I'm a little bit stuck, so any help will be appreciated. PROBLEM Find the complex Fourier coefficients of: $$f(t) = \sin(2\pi t)$$ and $$f(t) = |\sin(2\pi ...
-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 ...
2
votes
1answer
43 views

Equidistribution of $\{\xi_n\}$ where $\xi_n = <n\frac{p}{q}>$ $p,q$ rel. prime

I'm working from Stein's An Introduction to Fourier Analysis, and there's a question (chapter 4 number 6): Let $\theta = \frac{p}{q} \in \mathbb{Q}$ where $\operatorname{gcd}(p,q) = 1$. Assume ...
0
votes
0answers
19 views

Determine whether the set is uniqueness set

We say that $\Lambda$ is a uniqueness set for the Paley-Wiener space $PW_{\pi}$ if $$(F \in PW_{\pi} \wedge F|_{\Lambda}\equiv 0) \rightarrow F\equiv 0.$$ For example, $\Lambda =\mathbb Z$ is a ...
0
votes
1answer
40 views

$f(x) =\cos(x-y) -\cos(\delta)$ plotting

Ok, so this is a confusing one. I'm not sure what my teacher is looking for. The problem is: Plot any number $-\pi < y< \pi$ and pick a small number $\delta > 0$ so that the whole interval ...
1
vote
1answer
33 views

Proving uniform convergence with some kernel

Question Given $K_n=\cases 0$ elsewhere , $n- n^2|x|$ for $x<|\frac 1n|$ , $f$ is continuous, $2\pi$ periodic $\Bbb R \to \Bbb C$ . $f_n(x)=\int _{-\pi}^ \pi f(t)Kn(x-t)$ prove that ...
0
votes
1answer
47 views

Fourier series verification

Question: $$f(x)= \sum_{n=0}^\infty \frac {e^{inx}}{1+n^2}$$ if $x\ne 2\pi k$ and $f(x)=0$ if $x=0 , x=2\pi k$ Find $\hat f(n)$ Find the Fourier series of $\displaystyle g(x)=\int _0^xf(t)dt$ ...
0
votes
1answer
45 views

Discrete fourier transform problem

We have taken $1000$ observations from signal $s(t),t \in\Bbb{R}$ $$h(k)=s(k\Delta t+t_0),k=0,1,\dots,999,$$ where $\Delta t=1/200 $ and $t_0=-2$ (in seconds). When we calculate discrete fourier ...
0
votes
1answer
41 views

Convergence of a fourier series of $f(x)=1+\sin \frac {\pi^2}x$

Question: let $f:\Bbb R \to \Bbb R, f(0)=1 \forall x\in[-\pi,\pi] \setminus \lbrace0\rbrace , f(x)=1+\sin \frac {\pi^2}x$ Does the fourier series of this function converge at zero? If it does what is ...
1
vote
1answer
903 views

Fourier series coefficients proof

Can somebody help me understanding the fouries series coefficients? I know that if we have: $$f(n) = \sum_{n=1}^N A_n \sin(2\pi nt + Ph_n) \tag{where $Ph_n$ = phase}$$ And because of the ...
2
votes
1answer
68 views

Gibbs phenomenon in simple square wave

Given the square-wave function (later used to illustrate Gibbs phenomenon) $$f(x) = \left\{\begin{array}{c} \frac{h}{2} & 0 < x < \pi \\ -\frac{h}{2} & -\pi < x < 0 ...
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: ...
2
votes
2answers
157 views

Show a Fourier series converges uniformly

I need to show that the Fourier Series of |x| in the interval $(-\pi, \pi)$ converges uniformly to |x| in $[-\pi, \pi]$. I know that |x| = $\frac{\pi}{2}$ + ...
0
votes
1answer
104 views

Is the matrix Wn from the DFT a Hermitian operator?

A homework question asks me whether or not the matrix $W_N$ from the matrix representation of the Direct Fourier Transform is a Hermitian operator. From what I understand an Hermitian operator does ...
0
votes
1answer
120 views

Multi layer perceptron activation function

How can you show that the Fourier series approximation of a function (so $f(x)=\sum\limits_{n=0}^{\infty} (a_n cos(nx) + b_n sin(nx))$ can be approximated to arbitrary precision by a feedforward ...
2
votes
0answers
53 views

Fourier's Method Question

I've been asked to use Fourier's method to obtain the following solution; $$u(x,t) = \sum_{n=1}^{\infty} B_n e^{-(n \pi C / L)^2 t} \sin(\frac{n \pi x}{L})$$ $$B_n = \frac{2}{L} \int_0^L \sin(\frac{n ...
0
votes
0answers
100 views

Find the Fourier series of square wave graph.

Find the Fourier series of f(t) whose graph is given below. Is it correct to write it as a piece wise function given below then find $a_o$, $a_n$ and $b_n$ $f(x)=\begin{cases} -1& \text{-2 ...
0
votes
1answer
263 views

Find the Fourier series S(t) of the period 2$\pi$

Find the Fourier series S(t) of the period 2 function $f(t)=\begin{cases} -1& \text{if −$\pi$ < t < 0;}\\ \;\;\;1& \text{if $\:$0 < t < $\pi$;}\\ 0&\text{if $t = −\pi, 0, ...
2
votes
2answers
129 views

Sum of Fourier Series

I need to find the Fourier Series for $f\in \mathcal{C}_{st}$ that is given by $$f(x)=\begin{cases}0,\quad-\pi<x\le 0\\ \cos(x),\quad0\le x<\pi\end{cases}.$$ in the interval $]-\pi,\pi[$ ...
1
vote
1answer
80 views

These questions are all about Fourier analysis.

Please prove these equalities,these questions appear in the chapter of Fourier series. If you can use other methods,please tell me more about it, and I am glad to know how to solve the questions: ...
1
vote
1answer
197 views

Fourier Series of $f$ on the given interval

my goal is to find the Fourier series of f on the given interval: $$f(x) = \begin{cases} 0, & \text{if } -\pi < x < 0 \\ \sin(x), & \text{if } 0 \le x < \pi \end{cases}$$ I know ...
1
vote
1answer
233 views

Even and odd functions using integrals

If $f: [-r, r] \to\mathbb{R}$ is an even function, show that $g(x) = \cos(nx)$ is an even function and $h(x) = \sin(nx)$ is an odd function. Consider: $\int_{-r}^{r} f(x)\cos(nx)dx = 2\int_{0}^{r} ...
1
vote
1answer
62 views

Fourier series convergence in $L^2$

Consider a function $g \in L^2(-\pi,\pi)$ such that it is continuous at $x \in (-\pi,\pi)$. Prove that if the Fourier series of g converges at x then that implies g(x) is its limit. I was thinking ...
1
vote
0answers
59 views

Looking for feedback on Taylor Maclaurin and Fourier series

The Problem: You encounter the following wave when examining a digital switching circuit. You need to create a mathematical model so that you can examine changes in the circuit’s behavior. ...
0
votes
1answer
31 views

Fourier series identity

I need to prove that $\dfrac{a \sin(bx)}{1 - 2a \cos(bx) + a^2} = \sum_{n=1}^\infty a^n \sin(nbx)$ where $|a| < 1$. It seems that this can be proved by using Euler's formula identities for ...
4
votes
3answers
475 views

Finding the Fourier Series of $\sin(x)^2\cos(x)^3$

I'm currently struggling at calculation the Fourier series of the given function $$\sin(x)^2 \cos(x)^3$$ Given Euler's identity, I thought that using the exponential approach would be the easiest ...
5
votes
2answers
404 views

A Fourier series exercise

Can anyone give me a hand with this exercise about Fourier series? Let $f(x)=-\log|2\sin(\frac{x}{2})|\,\,\,$ $0\lt|x|\leq\pi$ 1) Prove that f is integrable in $[-\pi,\pi]$. 2) Calculate the ...
18
votes
1answer
519 views

Seeking Fourier series solution on Laplace equation…still looking, am I on track?

Okay, I've been working at this a couple of days now, I will try to give relevant details but will omit some intermediate steps. The problem as given says: Consider the BVP for $u=u(x,y)$: ...
2
votes
1answer
72 views

Computation of standard series

I am stuck on the computation of the following sum: $\sum_{k=1}^\infty e^{-n^2}\cos(n)$. Simple tricks fail and also i have no idea how to fit it for Fourier series. Are there any other ways?
0
votes
1answer
212 views

Exponential Form of Fourier Series

Problem Suppose $f$ is a continuous function on interval $[-\pi,\pi]$ such that $\sum_{n\in\mathbb{Z}} |c_n| < \infty$ where $c_n = \dfrac {1}{2\pi} \int_{-\pi}^\pi f(x)\cdot \exp(-inx)~dx$, the ...
3
votes
1answer
48 views

How to calculate $\sum_{k=1}^{\infty}\frac{1}{(a^2+k^2)^2}$ after calculating $\sum_{k=1}^{\infty}\frac{1}{a^2+k^2}$ using Parseval identity?

The task is to calculate sum $\sum_{k=1}^{\infty}\frac{1}{a^2+k^2}$ using Fourier coefficients of $f(x)=e^{ax}$. First of all I calculated Fourier coefficients of the sum: ...
0
votes
1answer
123 views

Function as a convolution product of other two

I need help with this: I have to prove that a function $f\in L_{2}(T)$ can be expressed as $f=g*h$ (convolution product) for some functions $g,h\in L_{2}(T)$ if and only if $(\hat{f}(n))_{n}\in ...
1
vote
2answers
435 views

Fourier coefficients of the product of two functions

Given two functions $f,g\in L^2(\mathbb{T})$, I have to prove that the Fourier coefficients of $fg$ are given by $$\hat{fg}(n)=\sum_{k\in{Z}}\hat{f}(n-k)\hat{g}(k)$$ and that this series converges ...