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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22 views

After calculating Fourier series coefficients for $x(t)=2 cos(4t) + 4 sin(10t)$, why am I getting all zeroes for all coefficients?

I am trying to find the Cosine/Sine Fourier series coefficients for the given equation: $$x(t)=2\cos(4t) + 4\sin(10t)$$ $\cos(4t)$ has a period of $T=\frac{\pi}{2}$, and $\sin(10t)$ has a period of ...
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1answer
16 views

Integral equality $\int_{-\pi}^\pi\dots = \int_{|t|\le \delta}\dots+\int_{\delta\le |t|\le \pi}\dots$

This is an excerpt from here (page 6, bottom) I don't know if this is a typo or not, but what exactly happened to the integral of $\int_{-\pi}^{-\delta}$ for the $|\sigma_n(x) - f(x)|$? I don't ...
3
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1answer
34 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 ...
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0answers
3 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
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1answer
31 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 ...
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0answers
15 views

Fourier transform of $|x|^{-s}$

Using the definition of Fourier transform $\hat{f}(p) = (2\pi)^{-n/2} \int_{\mathbb{R}^n} f(x) e^{ix \cdot p} \ dx$ where $u \in \mathbb{R}^n$. What is the fourier transform of $|x|^{-s}$.
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21 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 ...
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1answer
25 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 ...
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0answers
10 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.
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1answer
19 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
53 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 ...
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2answers
40 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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14 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 ...
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1answer
16 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
33 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
15 views

Fourier Series Transform of Full-Wave Rectifier

A first step in converting AC-power from the power-grid to the DC-power that most devices need is to utilize a full-wave rectifier. A full-wave rectifier converts a sinusoidal input (Asin(omegaot) ...
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0answers
27 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 ...
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16 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
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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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16 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= ...
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1answer
17 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 ...
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25 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 $a_0 (f) = 0$. ...
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1answer
47 views

what is the Fourier transform of this function $ xe^{-x} $ [closed]

what is the fourier transform of this function $$f(x)=\begin{cases} xe^{-x} \textrm{ if } x>0,\\ 0,\textrm{otherwise}. \end{cases}$$
2
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1answer
15 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
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1answer
20 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
21 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
13 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
46 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
11 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
31 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
34 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
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1answer
37 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
11 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
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2answers
40 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
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2answers
14 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
19 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 ...
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0answers
13 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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20 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
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1answer
50 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
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1answer
15 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?
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43 views

Using the Fourier series of $x^2$ to show $\sum_{n=1}^{\infty}\frac{1}{n^2} = \frac{\pi^2}{6}$? [closed]

If the Fourier series of $f(x) = x^2$ from $0 < x < 2\pi$ is given to be $$\displaystyle\frac{4\pi^2}{3} + \sum_{n=1}^{\infty}\left(\frac{4\cos(nx)}{n^2}-\frac{4\pi \sin(nx)}{n}\right)$$ how ...
13
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1answer
188 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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23 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} ...
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1answer
61 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) = ...
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1answer
14 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
59 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 ...
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0answers
38 views

Did I obtain the right solution for the Fourier series of the following problem

If a Fourier series is defined by $f(x) = \displaystyle\frac{A_{0}}{2} + \displaystyle\sum_{n=1}^{\infty}A_{n}\cos \left(\frac{n\pi x}{L}\right) + B_{n}\sin \left(\frac{n\pi x}{L} \right)$ then if ...
1
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1answer
73 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 $(f,g):= \int_{0}^{2\pi}fg dx$ ...
2
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1answer
50 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) = ...