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

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Uniqueness of solution for seperation of variables solvable PDEs

I am taking first course in PDEs and the only way i know of solving PDEs is separation of variables , and all the equations i saw had unique answers due to the ICs and BCs , but not this one : $$ ...
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1answer
257 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
30 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 ...
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1answer
90 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 ...
3
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1answer
71 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
32 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
41 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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1answer
41 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 ...
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0answers
2k 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
93 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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1answer
33 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
47 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
39 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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1answer
21 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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1answer
44 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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1answer
624 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
24 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
33 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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2answers
61 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
33 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 ...
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0answers
46 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 ...
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1answer
174 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
50 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 < ...
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3answers
75 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 ...
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2answers
31 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
55 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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1answer
67 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, ...
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1answer
318 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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1answer
69 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
33 views

How to write a fourier series using periodic boundary conditions

Would writing $$ f(x) = x^2 $$ as a Fourier series using periodic boundary conditions on $-L < x < L$ with a basis of $$ e^{\frac{i\pi nx}{L}} $$ be just \begin{align}\bigl\langle ...
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1answer
138 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 ...
2
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1answer
140 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 $(f,g):= \int_{0}^{2\pi}fg dx$ ...
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1answer
72 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) = ...
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1answer
66 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 ...
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2answers
27 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
42 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 ...
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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 ...
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2answers
204 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 ...
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1answer
28 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, ...
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191 views

Proof: $f$ square-integrable $\Rightarrow f$ absolutely integrable on $[0, 2\pi]$

In a book I found the following statement: Let $\varphi(x)$ and $\psi(x)$ be square integrable, then $|\varphi \psi| \leq \frac{1}{2} |\varphi^2 + \psi^2|$. This implies, that every square ...
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1answer
85 views

If a continuous function on $[0,\pi]$ integrates to zero against cosines, it is identically constant

Let $f(x)$ is Continuous function on $[0,\pi]$,and for $n=1,2,.....,$ the function $f(x)$ has the following property:$$\int_{0}^{\pi}f(x)\cos{(nx)}dx=0.(n=1,2,......)$$ Proof: $f(x)\equiv C$(C is ...
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1answer
33 views

fourier series for g(x)=x between -pi and pi

Consider the following function defined on a finite interval: $$g(x) = x, 0\leq x\leq \pi $$ (3) (a) Sketch an even periodic extension of g(x). (b) Show that the Fourier cosine series representation ...
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2answers
47 views

Fourier series, instantly determining $b_n$ once $a_n$ is found.

Find the Fourier series of the following function: $f(x) = \left\{\begin{align} 1+x,\quad -1\lt x \lt 0 \\ 1-x,\;\;\;\quad 0\lt x \lt 1\end{align} \right.$ $f(x+2) = f(x),\quad\quad -\infty \lt x ...
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1answer
102 views

How did Fourier series lead to the development of rigorous analysis?

Once I've heard that the studies of Fourier series have lead to rigorous definitions of such concepts as function, convergence, integral, limit. And also that Cantor's study of Fourier series led him ...
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1answer
37 views

Fourier Series Coefficient Question

In calculating the Fourier Coefficients a0, an, bn: Why are the an and bn coefficients integrated over 2 times the inverse of the period, 2(1/T) while the a0 coefficient is integrated only over one ...
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1answer
303 views

Showing a series is not the fourier series of a riemann integrable function.

I want to show that the series $\sum_1^\infty \frac{sin(nx)}{\sqrt{n}}$ is not the Fourier series of a Riemann integrable function on $[-\pi,\pi]$. I was going to do this by showing that the partial ...
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1answer
43 views

Fourier series representation of $\sin^4 x$

I tried solving for fourier coefficients of Fourier series for the multiples of fundamental frequency $\omega_0=2$. So $F_n=\int_0^{\pi} \sin^4 x \, e^{-i2nx} dx$. And my calculator says answer should ...
3
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1answer
259 views

Weighted sum of cosines

Consider $$f(x) = \sum_{k=1}^\infty \cos(kx) k^\alpha.$$ The first question is: does this have a name (Mathematica gives it as a sum of polylogs of complex arguments, but this seems unnatural). Also, ...
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1answer
44 views

Fejer's theorem with Riemann integrable function

If $f$ is integrable and $f(x+), f(x-)$ exists for some $x$, then $$ \lim_{N \rightarrow \infty} {\frac{1}{{2\pi }}\int_{ - \pi }^\pi {f\left( {x - t} \right){K_N}\left( t \right)dt} } = ...
0
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1answer
32 views

Express as a complex Fourier series

My function is $f(x)= \dfrac{1}{1-2e^{ix}} + \dfrac{1}{1-2e^{-ix}} $, which has been periodically extended by $2\pi$. I found $C_0$ to be $\pi$. I'm having trouble expressing $C_n$. All I have is ...