Questions on Fourier series, the expansion of a function in terms of basis functions that satisfy an orthogonality relation. Usually, complex exponentials/sines and cosines are used.

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3
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1answer
52 views

Prove $\cos x = \frac{8}{\pi}\sum_n \frac{n\sin 2nx}{4n^2-1}$ with Fourier series

I want to prove $$\cos x = \frac{8}{\pi}\sum_n \frac{n\sin 2nx}{4n^2-1}\;x\in(0,2\pi)\;\;\;\;[1]$$ I have two questions regarding this: $(1)$ How can I find a function $f$ such that the former ...
0
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0answers
8 views

Short-Time-Fourier-Transform: why overlapping the window?

For STFT, we impose window of certain size onto the original signal, then we perform fft on each window. The uncertanty about frequency and time is determined by the width of the window, however, I ...
3
votes
1answer
43 views

Fourier series for $f(x)=\begin{cases} 0 & -\pi\leq x<0 \\\sin x & 0\leq x\leq \pi \end{cases}$

Find the Fourier series for $$f(x)=\begin{cases} 0 & -\pi\leq x<0 \\\sin x & 0\leq x\leq \pi \end{cases}$$ I found an answer, I'm not completly sure if it's right. The solution would ...
3
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0answers
24 views

Local behavior of a Fourier series and a intgral

So I have to calculate an integral that involves a Fourier series of some function. I would like to get some kind of local control of the function near zero the series is ...
1
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1answer
31 views

Prove that the toral endomorphism $T_A: \mathbb{R}^d / \mathbb{Z}^d \rightarrow \mathbb{R}^d / \mathbb{Z}^d$ is not ergodic.

Prove that the toral endomorphism $T_A: \mathbb{R}^d / \mathbb{Z}^d \rightarrow \mathbb{R}^d / \mathbb{Z}^d$ is not ergodic. A is an integer matrix such that A has an eigenvalue which is a ...
0
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0answers
13 views

How to determine singularities of a series?

Given a double Fourier series, how do we determine its singularities ? PS: I wonder how we find singularities(mathematically) if a function cannot be expressed in a closed form.
2
votes
1answer
24 views

Fourier Series Coefficient

I am trying to review the basics. Find the Fourier series for the function $$f(x) =\left\{ \begin{array}{l l} 2x & \quad -\frac{\pi}{2}<x<\frac{\pi}{2}\\ 0 & \quad ...
2
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0answers
109 views

How to find the Total Harmonic Distortion of a Periodic Signal through MATLAB?

How to find the Total Harmonic Distortion of a Periodic Signal through MATLAB? I just need help in confirming if my way of approach to finding the THD seems valid, I'm new to MATLAB so I'm not quite ...
1
vote
2answers
47 views

Finding Fourier cosine series of sine function

I am trying to find Fourier cosine series of following function, but think that I am messing up somewhere. $$ f(x)=\sin \bigg ( \frac{\pi x}{l} \bigg ) $$ Fourier cosine series can be written as $$ ...
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0answers
20 views

Converting sum of complex exponential to sum of cosine

So I am trying to convert the equation $$\sum_{k=-2}^2 \alpha_k e^{i \frac{2 \pi}{T_0} kt}$$ Where $\alpha_0 = 1$, $\alpha_1 = 2 \angle30^\circ$, $\alpha_{-1} = 2 \angle{-30^\circ}$, $\alpha_2 = 1 ...
0
votes
1answer
18 views

For $f(\theta)= e^{\theta}$. Is it true that $\hat{f}(n)(1-in)=0$ for all $n\in \mathbb Z.$?

(This is motivated from the following question) Fact: If $f \in C^1(\mathbb{T})$, then the Fourier coefficients $\widehat{f'}(n)$ of the derivative $f′$ can be expressed in terms of the Fourier ...
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0answers
35 views

Solving the PDE $\frac{\partial u}{\partial t}=a\frac{\partial^2 u}{\partial x^2}+b\frac{\partial u}{\partial x}$

I am trying to solve the PDE $\frac{\partial u}{\partial t}=a\frac{\partial^2 u}{\partial x^2}+b\frac{\partial u}{\partial x}$ for constants $a$ and $b$ with conditions $\frac{\partial u}{\partial ...
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0answers
28 views

A differential equation regarding Fourier series. [Updated]

Define $PC^r(2\pi)=\{f:[-\pi,\pi]\rightarrow\mathbb{R}: f\in\mathcal{C}^r \text{ and } f,f',f'',\dots, f^{(r)} \text{ are } 2\pi\text{-periodic} \}$. I want to show that if $g\in PC^1(2\pi)$ and $f\in ...
1
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0answers
24 views

Bound of a fourier series when coefficients are bounded

Let $f(x)$ be a finite fourier series with $$f(x)=a_0+\sum_{n=1}^N\left(a_n\sin{\left(2\pi nx/P\right)}+b_n\cos{\left(2\pi nx/P\right)}\right)$$ and bounded coefficients ...
1
vote
1answer
18 views

Periodic Functions of Cycle greater than two.

I am now aware of periodic functions, and how they cycle like binary flags. It helps to use this in discrete math. This brings me to my problem. I am trying to make a function that is periodic for ...
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0answers
54 views

Solution of Laplace equation for a regular hexagon

I am trying to analytically solve Laplace equation in a regular hexagon. My equation is $\nabla^2 \phi=0$ and boundary conditions are : $\phi = 0$ (at the base of hexagon) $\phi= \phi_1$ $\phi= ...
0
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1answer
20 views

Vibration on a rectangular Plate

I am trying to solve a problem that has been set for me. I haven't come across a problem like this like, so i need some help getting through it. It is used to model the vibrations of a rectangular ...
0
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0answers
34 views

On Wirtinger's inequality for functions without continuity of the derivative, is this proof correct?

If $f$ is a $2\pi-$periodic, piecewise continuous function such that $f'$ is also a $2\pi-$periodic piecewise continuous function. Suppose that the first Fourier coefficient of $f$ is $a_0 (f) = 0$. ...
2
votes
1answer
22 views

fourier series of absolute value of function

I am trying to find the Fourier series of $$ |\cos(x)| \text{ from } -\pi \leq x<\pi$$ I know that the $$ b_n $$ terms go to 0 because we have the integrand as an odd function of x. But how can ...
2
votes
1answer
21 views

Determining if two expressions are equal, in order to ensure a Fourier series is correct

Motivation: I have a question that asked me to find the Fourier series of some function $f(x) = \left\{\begin{array}A,\quad -1\lt x \leq 0 \\ Ax, \quad 0 \lt x \leq 1 \end{array}\right.$ periodic on ...
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0answers
27 views

Basis for quaternionic functions

We know that the set of functions $\{1,\cos x, \sin x, \cos 2x, \sin 2x, ... \; | \,x \in \mathbb{R} \}$ is a basis in the space $L^2_\mathbb{R}[-\pi,\pi]$ . Given a quaternion $z \in \mathbb{H}$ ...
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0answers
17 views

Fourier sine series

Let the function given by $f(x)=\cos(x)-1+ \dfrac{2x}{\pi}$ defined on the interval $[0,\pi]$. a) Calculates the development of Fourier sine series. b) Study the uniform convergence on $[0,\pi]$ the ...
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2answers
51 views

Showing two things are equal by Fourier series

Given the Fourier series for the function: $$f(x) = x+\frac14x^2 \quad -\pi\leq x \lt \pi$$ $$f(x)=f(x+2\pi) \quad -\infty \leq x \lt \infty$$ is $$\frac{\pi^2}{12}+\sum \limits_{n=1}^\infty (-1)^n ...
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0answers
16 views

How to solve an inhomogenous PDE using Fourier Transform

$u_{tt}=u_{xx}+(8-64x^2)e^{-4x^2}$ $u(x,0)=e^{-4x^2},u_t(t,0)=0$ $0<t<\infty,-\infty<x<\infty$ By Fourier Transform ...
2
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0answers
37 views

Accessible textbook about basic Fourier analysis in terms of integrals wrt measures

I am looking for a basic and accessible textbook (or set of lecture notes) that discusses basic fourier analysis but in terms of measures and integrals with respect to measures. Not sure if it is done ...
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0answers
38 views

A question on convergence of derivative of power series

This is a question from Fourier Analysis with Applications by Folland. First we write Fourier series for $$e^{\theta}=\sum c_ne^{in\theta}$$ We differentiate this series term by term to obtain ...
1
vote
1answer
39 views

Fourier Series of Real-valued Functions

Context: For a $2\pi$-periodic bounded function $f:\mathbb{R}\to\mathbb{C}$, we define the complex Fourier coefficients of $f$ by $$ \hat{f_k}:=\frac{1}{2\pi}\int_0^{2\pi}f(x)e^{-ikx}\,dx. $$ We call ...
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0answers
22 views

Fourier Cosine series expansion for two dimensional function

I have a two dimensional function with its values and range. I need to expand the function in Fourier cosine series. The function as follows: $$f(x,y) = \begin{cases} A &, -\frac{L}{2} + 2nL < ...
0
votes
2answers
43 views

Fourier series of $f(x)=1$

$\displaystyle f(x)=\frac{a_{0}}{2}+\sum_{n=1}^\infty a_{n}\cos nx$, where $a_{n}=\frac{2}{\pi}\int_0^\pi f(t)\cos(nt) \ dt$, if $f$ is even. But for $f(x)=1$, the left side goes to $0$. How can I ...
0
votes
2answers
16 views

Using Weistrass Approximation Theorem to define fourier series convergence.

Weistrass Approximation Theorem: Let f be continuous on [-$\pi$,$\pi$] with $f(-\pi)=f(\pi)$. Then for each $\epsilon>0$ there is a trigonometric polynomial T such that $|f(x)-T(x)|<\epsilon$ ...
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0answers
24 views

Convergence of Fourier Sine Series for Gerneral Continuous Function

This is my question: How do I should that, for $f \in C[0,\pi]$ with $f(0) = f(\pi) = 0$, the Fourier sine series $$\tilde f_n = \sum_{r=0}^n b_r \sin(r s)$$ converges uniformly to $f$ on ...
0
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0answers
15 views

holder cts then the fourier coefficient absolutely convergence

I know it is the bernstein theorem, but it seems too complicated,do we have some simple mothod to solve it??
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0answers
23 views

Find Fourier expansion of $g(x)$ and then deduce that$\frac{\pi ^{2}}{8}=1+\frac{1}{2^{3}}+\frac{1}{5^{3}}+\frac{1}{7^{3}}+…$

Given $g(x)=\left\{\begin{matrix} \, \, \, \, \, \, 0\ if\; -\pi < x\leqslant 0\\ x\ if\: 0\leq x\leq \pi \end{matrix}\right.$ find the Fourier expansion and deduce that: $\frac{\pi ...
2
votes
1answer
57 views

Express $f(x)=\sin{x}$ as an even function

Express $f(x)=\sin{(x)}$, with $(0 < x< \pi )$ as an even function, $f(x+ 2\pi)=f(x)$ The topic is on Fourier Series. I have the following so far: Since $f(x)$ must be an even function, ...
0
votes
1answer
50 views

CTFT and DTFT in MATLAB

Is it possible to plot CTFT and DTFT in MATLAB? I know of DFTs(FFTs) in MATLAB since I am using them but what if I want CTFT and DTFT? If yes, then what function shoulf I use?
13
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1answer
233 views

Integral that arises from the derivation of Kummer's Fourier expansion of $\ln{\Gamma(x)}$

I am trying to prove that for $0<x<1$, $$\color{blue}{\ln{\Gamma(x)}=\frac{1}{2}\ln(2\pi)+\sum^\infty_{n=1}\left\{\frac{1}{2n}\cos(2\pi nx)+\frac{\gamma+\ln(2\pi n)}{n\pi}\sin(2\pi ...
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0answers
24 views

Did I set up my integrals right in preparing to find a Fourier series?

If you take the Fourier series of a function $f(x)$ where $0 < x < \pi$, then would $a_{0}$, $a_{n}$, and $b_{n}$ be defined as, $a_{0} = \displaystyle\frac{1}{\pi}\int_{0}^{\pi}f(x)dx$ $a_{n} ...
0
votes
1answer
67 views

Fourier series of $f(x) = x - [x]$, where $[x] = n \in Z$ s.t. $n \leq x < n+1$

How do you find the Fourier series of $f(x) = x - [x]$, where $[x] = n \in Z$ s.t. $n \leq x < n+1$? I am familiar with Fourier series and use the following definition to solve them: $f(x) = ...
1
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1answer
18 views

How to write a fourier series using periodic boundary conditions

Would writing \begin{align}x^{2}\end{align} as a Fourier series using periodic boundary conditions on -L < x < L with a basis of \begin{align}e^{\frac{i\pi nx}{L}}\end{align} be just ...
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0answers
23 views

Two definition of Fourier's transformation agrees? [duplicate]

Definition 1: If $f\in L^1(R^n)$, $\hat{f} (s)=\int _{R^n} e^{-isx}f(x)dx$ Definition 2: If $f\in L^2(R^n)$, let $f_i \in$ {Schwartz functions} such that $f_i$ converges to $f$ in $L^2$, then ...
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1answer
70 views

Solving $\sum_{n=1}^{\infty} \frac{1}{n^2}$ using the fourier series.

Please do NOT solve the problem, I just need some help, not a full solution. I would like to try this myself. Find $\zeta(2) = \displaystyle \sum_{n=1}^{\infty} \frac{1}{n^2}$ The fourier series for ...
1
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1answer
90 views

Questions about Fourier Series

I have recently started looking at the topoic of Fourier series. Consider the space of square integrable functions $L_{2}[0,2\pi]$. Where we define the inner product as $(f,g):= \int_{0}^{2\pi}fg dx$ ...
3
votes
1answer
60 views

Did I calculate this Fourier series correctly?

If we use the definition of the Fourier series in the following way: $$f(x) = \frac{A_{0}}{2} + \sum_{n=1}^\infty A_n \cos(nx) + B_n \sin(nx)$$ then if $-\pi < x < \pi$, and $f(x) = ...
0
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1answer
38 views

Helpful Integrals for evaluating Fourier series, my book is wrong?

I don't understand why my book is claiming the following for any $n$ or $w_0$ this is always the case over one period. I think it depends on the $w_o$ really. I have proof too, but I just want another ...
1
vote
2answers
26 views

Help with setting up the Fourier series for the following functions.

i. $f(x) = \operatorname{sgn}(x)$ for $-\pi < x < \pi$ where $$\operatorname{sgn}(x) = \begin{cases} 1, & x>0, \\ 0, & x=0, \\ -1, & x<0. \end{cases} $$ ii. $f(x) = ...
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0answers
16 views

Finding maxima of periodic function with Fourier series

I have a periodic function for which I have computed the Fourier series coefficients (98 terms). For this periodic function I need to find the maximum value. Is there an analytical function that gives ...
3
votes
0answers
33 views

Discrete Fourier Transform of a shift of a tuple over a finite field

Let $a = a_0 a_1 \cdots a_{N-1}$ be a sequence over a finite field $\mathbb{F}_q$, where $N \mid q^n-1$ for some $n$. Let $\xi_N$ be a primitive $N$-th root of unity in the extension ...
5
votes
0answers
47 views

Accelerated Eigenfunction Expansions of Random Functions

I am interested in eigenfunction expansions of random functions. We know that the autocorrelation of brownian motion, $\{ B_t \}_{t \geq 0}$, is given by $$ E[B_t B_s] = \min\{s,t \}, $$ which can ...
3
votes
2answers
180 views

Different Versions of Fourier Series? What about Uniqueness?

Let $f(x)$ be a function, then for its Fourier series $$ f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos(nx) + b_n \sin(nx)) $$ I found two different definitions (both yielding different ...
0
votes
1answer
24 views

write y(x) according to sin(d+(ay+b)/cx)=y

I end up with the formula $\sin (d+\frac{(ay+b)}{cx})=y $, and try to write $y$ as a function of $x$. There can be multiple solutions (of $y)$ to $\sin(d+\frac{(ay+b)}{cx})=y$ pretending $x$ is known, ...