0
votes
0answers
19 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
31 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
40 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
43 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
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 ...
2
votes
1answer
34 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
18 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
34 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
25 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
44 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
36 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
35 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
580 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
51 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
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
523 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
116 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
71 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
107 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
47 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
86 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
261 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
123 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
79 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
129 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
204 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
59 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
52 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
30 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
449 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
375 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
497 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
203 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
47 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
100 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
336 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 ...
2
votes
3answers
187 views

Calculating the Fourier series of $x^{3}$

I was given as homework to calculate the Fourier series of $x^{3}$. I know, in general, how to obtain the coefficients of the series using integration with $$\sin(nx),\cos(nx)$$ multiplied by the ...
4
votes
1answer
118 views

Fourier series of a function

Consider $$ f(t)= \begin{cases} 1 \mbox{ ; } 0<t<1\\ 2-t \mbox{ ; } 1<t<2 \end{cases}$$ Let $f_1(t)$ be the Fourier sine series and $f_2(t)$ be the Fourier cosine series of $f$, ...
1
vote
2answers
153 views

Prove that if $\sum c_n e^{inx}$ converges in $L^2$ to $f$ then $c_n$ are the Fourier coefficients.

The full problem is this: Let $f:[-\pi,\pi]\rightarrow\mathbb{C}$ be Riemann integrable. Let $\{c_n\}_{n\in\mathbb{Z}}\subset\mathbb{C}$. Prove that if $s_N=\sum_{n=-N}^Nc_ne^{inx}$ converges in ...
4
votes
0answers
99 views

Prove a function has $k$ continuous derivatives from its Fourier series

Here is the problem. Let $k\in \mathbb{N}$. Suppose that there is a constant $C$ such that $|c_n|<\frac{C}{|n|^{k+1}}$ ($c_n$ here is the $n$th Fourier coefficient). Prove that ...
1
vote
1answer
77 views

$\int_{-\infty}^{\infty}f(\xi)d\xi = \lim_{\delta \to 0 }\sum_{n=-\infty}^{\infty}\delta f(\delta n)$?

Assume that $f$ is continuous and moderate decrease, show that $$\int_{-\infty}^{\infty}f(\xi)d\xi = \lim_{\delta \to 0,~ \delta>0}\sum_{n=-\infty}^{\infty}\delta f(\delta n)$$ From the ...
2
votes
1answer
468 views

Fourier and Legendre series

Find the Fourier sin series for the function $f(x) = x^3$ on the interval $0\leq x \leq L$. the Legendre series for the same function. One representation involves an infinite number of terms, ...
1
vote
1answer
66 views

Let $\alpha$ be a real number, and let $g(\theta) = f(\theta - \alpha), \theta \in \mathbb{R}$. Find the Fourier series for g.

Suppose that $f(\theta)$ is a $2\pi$ periodic function with a known Fourier series. Let $\alpha$ be a real number, and let $g(\theta) = f(\theta - \alpha), \theta \in \mathbb{R}$. Find the Fourier ...
9
votes
7answers
464 views

Why does this Fourier series have a finite number of terms?

I am learning about Fourier series in class and the basic form of a Fourier Series is $$a_{0}+\sum_{n=1}^{\infty} [a_{n}\cos(nx)+b_{n}\sin(nx)]$$ so a fourier series should have an infinity number ...
0
votes
1answer
164 views

Confused about Fourier Series

I've been told to construct a Fourier Series for the odd function that has period $2\pi$ and is equal to $\cos(x)$ for $x \in (0,\pi]$. For $f$ that is $2\pi$ period I have a formula ...
1
vote
1answer
155 views

Sine series of $\pi/2$

I'm studying Fourier series and came across this peculiar problem. I just studied (along with proper reasoning) that if $f(x)$ is an even function, then the fourier series has only Cosine terms and if ...
2
votes
2answers
98 views

The difference between m and n in calculating a Fourier series

I am studying for an exam in Differential Equations, and one of the topics I should know about is Fourier series. Now, I am using Boyce 9e, and in there I found the general equation for a Fourier ...
0
votes
2answers
383 views

How to integrate $e^{r\cos x} \cos(r\sin x)$

The title says everything. I'm studying fourier series and I've stumbled upon this question: find the fourier series of $f(x) = e^{r\cos x} \cos(r\sin x)$. So that i need to integrate this function ...