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Questions tagged [fourier-series]

A Fourier series is a decomposition of a periodic function as a linear combination of sines and cosines, or complex exponentials.

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Fourier( complex Number included) plot in Matlab

How can I plot this fourier transform as an amplitude spectrum in Matlab? I can plot in Matlab, but for this function I do not know how the complex function will affect my plot? Thank you $$F(w)=(10/...
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Find the phase angle form of the Fourier series of:

$f ( a ) = 3 a ^ { 2 } \quad$ where: $0 \leq a < 4$ and $f ( a + 4 ) = f ( a )$ My Solution $f ( a ) = 3 a ^ { 2 }$ where $0 \leq a \leq 4$ Solution given $\int ( a + 4 ) = 3 ( a + 4 ) ^ { 2 }$ ...
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Find extrema and roots of a function from its Fourier series expansion

That might be a dumb question, but here it comes. Say we have a function defined by its fourier series expansion as follows : $$\forall x\in\mathbb{R}, f(x)=\sum\limits_{n=1}^{\infty}A_n\cos\left(2\...
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Evaluating Fourier coefficients to complete a Laplace equation solution

While solving a PDE problem involving the Laplace equation in 3D, I arrive at the following summation relation when i substitute the only non-homogeneous boundary condition available $$ \sum_{m=1}^{\...
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An Sobolev-type inequality of $1-d$ torus

In Kishimoto-Tsutsumi's paper published by Math Research Letter 2018, I see the following inequality in the last line of page 10: $\| f g \partial_x h \|_{H^{-s}(T)} \lesssim \| f \|_{H^s(T)} \| g \...
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Quasi-periodic sequence

Let $f(\theta)$ be some $2\pi$-periodic function which takes the values $f(\theta) \in \{1,-1\}$. Further let $Q$ be some number which is rationally independent of $2\pi$ (More specifically take $Q/(2\...
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When does Riemann-Lebesgue lemma hold in general?

Let's say for simplicity that I'm on the torus $\mathbb{T}=\mathbb{R}/\mathbb{Z}$. In this setting, Riemann-Lebesgue lemma could be stated as: for any $f\in L^1(\mathbb{T})$, $$ \lim_{\vert k\vert\to\...
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How can I find the Fourier series of $f(x) = x^2$ which is a $2\pi - periodic$ function on the interval $[0,2\pi)$

Find the Fourier series of $f(x) = x^2$ which is a $2\pi - periodic$ function on the interval $[0,2\pi)$ My question exactly: what is the difference between the solution of the above question and ...
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Fourier Series of Triangular waveform

Fourier Series of Triangular waveform this is the solution of Fourier series of a triangular waveform from the book Circuits and Networks: Analysis and Synthesis by Shyammohan S. Palli. In this ...
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Proving the uniform convergence of the Fourier series of $f(x) = x^2$

I calculted the Fourier series for $f(x)= x^2$ and I get that it is: $$\frac{ \pi^{2}}{3} + 4 \sum_{m = 1}^{\infty} \frac{(-1)^m}{m^2} \cos (mx).$$ But the rest of the question asks me to show ...
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Compute $\sum\limits_{n\in\mathbb{N}}\frac{1}{n^2}$ using Fourier series.

So yeah, these "compute sum by using the Fourier expansion" questions are starting to piss me off somewhat. I feel like If I have 50 of these questions in a row, I can never use the technique learned ...
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A clarification of Fourier representation of a function

As I was studying Fourier representation of a $T$-periodic function $f(t)$, I came up with two different definitions. For a function with discrete domain, $f(t) = \sum\limits_{k=0}^{T-1} C_k e^{i2\pi ...
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Use Fourier series of $f(x)=x(\pi-|x|)$ in $(-\pi,\pi)$ to compute the series $\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{(2n-1)^3}.$

The fourier series I obtained was $$x(\pi-|x|)=\frac{4}{\pi}\sum_{n\in\mathbb{N}}\frac{(-1)^{n+1}+1}{n^3}\sin(nx),\tag1$$ which I checked is correct by plotting RHS and LHS. I'm not sure what to set ...
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Compute $\sum_{n=1}^\infty\frac{(-1)^{n+1}}{4n^2-1}.$

So I need to use the fourier series of $f(x)=|\sin(x)|$ in $(-\pi,\pi)$ to compute the sums $$\sum_{n\in\mathbb{N}}\frac{1}{4n^2-1}\tag1,$$ $$\sum_{n\in\mathbb{N}}\frac{(-1)^{n+1}}{4n^2-1}.\tag2$$ ...
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Every countable closed set is of uniqueness

I am following the proof of Cantor's theorem that every countable closed set is of uniqueness as given in Kechris' notes (available here, thm. 4.2, proof on p. 12). My doubts are the following: how ...
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Solve Laplace's equation with boundary conditions

A problem from Fourier Analysis An Introduction. Chapter 2, 19: Solve Laplace's equation $\Delta u=0$ in the semi infinite strip $$S=\{(x,y):0<x<1,0<y\},$$ subject to the following ...
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Complex exponential series with different limits

Can anyone help in beginning the expansion to this? I know what the general expansion of a complex exponential series should look like and what Cn is equal to but because the limits to this function ...
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1answer
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Compute fourier series of $f(x)=e^{bx}$ on $(-\pi,\pi)$.

So I just applied the straightforward method by calculating $a_0,$ $a_n$ and $b_n$. I got that $$a_0=\frac{1}{\pi}\int\limits_{-\pi}^{\pi}e^{\pi b}dx=\frac{2\sinh(\pi b)}{\pi b}.\tag{1}$$ By ...
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Question about $a_0$ in the Fourier series for $f(x)=x^2$ on $(-\pi,\pi).$

So I solved this assignment and got the same answer as the person in this thread. The answer is correct and I also verified it by plotting with software. The answers are $$f(x)=\frac{\pi^2}{3}+4\sum_{...
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Confused on Fourier Transform

Currently I am reading a paper related to Computer Graphics. However, the core algorithm in the paper uses Fourier Transform. I learned Fourier Transform by myself and I still don't understand some ...
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A difficulty in understanding an example of a remark on pointwise convergence of Fourier series.

The theorem and the remark and an example on a remark are given below: But I do not understand the example $f(x) = x$, specifically I do not understand: 1- what do the author mean by "$f(x) = x$ as ...
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Best Approximation Theorem - Fourier Series

I'm reading Stein & Shakarchi's Fourier Analysis, but had a question about a small part of the proof of the mean-square convergence of fourier series of a function $f$. To show the convergence, ...
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Fourier coefficients calculator Apps or websites [closed]

Is there any platform that helps giving Fourier coefficients for a function that I give please ??? I need to check my answers
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Abel means of function f when having a jump discontinuity

Fourier Analysis An Introduction Chapter 2, Exercise 17 (a): Abel means of $f$ converge to $f$ whenever $f$ is continuous at $\theta$: $$\lim_{r\to1}A_{r}(f)(\theta)=\lim_{r\to1}(P_{r}*f)(\theta)=f(\...
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Fourier series for a piecewise defined function.

Compute the Fourier series of the function defined on $(-\pi,\pi),$ assuming $0<a<\pi,$ $$f(x)=\left\{ \begin{array}{rcr} 1 &-a<x<a& \\ -1 &2a<x<4a& ...
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Fourier series of $|\sin(x)|,$ confusing $a_0$ and $\frac{1}{2}a_0.$

So according to definition, the fourier series for an even function $f(x)$ is given by $$f(x)=a_0+\sum_{n\in N}a_n\cos\left(\frac{\pi n x}{L}\right)\tag1$$ where $$a_0=\frac{1}{2L}\int\limits_{-L}^{L}...
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Fourier series of $|\sin{x}|$ in $(-\pi,\pi),$ questions regarding $L.$

I'm tasked to find the Fourier series of $f(x)=|\sin{x}|$ in $(-\pi,\pi)$. The period of $f$ is clearly $\pi$ since $|\sin(x+\pi)|=|\sin(x)|.$ So we set $2L=\pi\Leftrightarrow L=\pi/2.$ Since this is ...
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Fourier representation of a function with changing period

I learned that the Fourier representation of a periodic function $f(t)$ with period $T$ is $\sum \limits_{k=-\infty}^{\infty} C_k e^{i2\pi k t/T}$ with appropriate constants $\{C_k\}$. My question: ...
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Convolution - references request

From which book this chapter is ? http://www.math.ncku.edu.tw/~rchen/2016%20Teaching/Chapter%203_Convolution.pdf
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Property of the Fourier Series

Suppose that $\sum_{k=-\infty}^{\infty}c_ke^{ikx}$ is a Fourier Series for $f(x)$. To what function does $\sum_{k=-\infty}^{\infty}c_ke^{i4kx}$ converge to? My best estimate is $f(4x)$ but I'm not ...
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Average of product of Fourier transform of two signals

I have two signals which depend on x and z, $a(x,z)$ and $b(x,z)$. Their Fourier transform along both directions is denoted as $A(k_x,k_z)$ and $B(k_x,k_z)$, respectively. I would like to compute the ...
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Why multiply by cosine on both sides in Fourier coefficient derivation?

I'm new to Fourier Series and I'm studying the derivation of the Series. Specifically, I'm looking at how to derive (e.g. solve for $a_n$) the coefficients in $f(x)=a_0 + \sum_{n=1}^{∞}{a_n cos(n\...
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Fourier series converge in $L^2$ Proof Explanation

I am using multiple textbooks on Fourier Series, and the only one that contains a proof is Mark A. Pinsky's Introduction to Fourier Analysis and Wavelets. However, since it is so concise, I am asking ...
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Fourier inversion formula in higher dimensions

Can I simplify the integral $$\displaystyle \int_{[-a,a]^n} e^{-i \langle x,\xi \rangle} \phi(|\xi|)d\xi$$ by means of the identity $\langle x,\xi \rangle=r \cos(\theta)$, with $r=|x||y|$? If yes, ...
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A difficulty in understanding the proof of Riemann Lebesgue lemma(2)

A part of the proof is below: 1-But it seems to me that the last line the last inequality is incorrect ..... am I correct? 2-Also if I am going to prove that the second limit equals zero I will use $...
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Calculating the Fourier series for the function $y = 1,-\pi \leq x \leq \pi$.

Calculating the Fourier series for the function $y = 1,-\pi \leq x \leq \pi$. My answer: I have calculated it and I got $a_{0} = 2, a_{m} = b_{m} = 0.$ so the Fourier series of 1 is 1. Am I correct?
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A difficulty in understanding the proof of Riemann Lebesgue lemma.

The proof is given below: My questions are: 1- In the second line from below how do we get $2/|\lambda|$ (in the second term) from the line before it. 2- What is the lemma trying to say?
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how to show cos($kx$) is orthogonal to sin($kx$)?

In $\mathscr L_2$, it's easy to prove (cos$x$, sin$x$, cos$2x$, sin$2x$...cos$Nx$, sin$Nx$) is an orthogonal basis. However, in $\mathscr L_2$, how to prove cos($kx$) is orthogonal to sin($kx$)? I am ...
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Proving that a given $u$ is a solution to $u_{tt} = c^2 u_{xx}.$

Proving that a given $u$ is a solution to $u_{tt} = c^2 u_{xx}.$ The question asks me to prove that $$u(x,t) = F(ct + x) + G(ct -x),$$ is a solution of the above equation on condition that we know ...
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Approximating Fourier transformed function with Fourier series

Given a function $f(x)$, composed with continuous frequencies in certain interval $[-m,m]$,\begin{equation} f(x) = \frac{1}{a}\int_{-m}^{+m} \hat{f}\Big(\frac{\xi}{a}\Big)e^{\frac{i2\pi x\xi} {a}}d\xi;...
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From $L^p$ to $L^q$ Norm for finite trigonometric polynomial

Let $1\leq p\leq q \leq \infty$ and let $f\in L^p([0,2\pi])$ such that $\hat{f}(k)=0$ for all $\vert k\vert > N$. I would like to show that $$\Vert f \Vert_p \leq (2N-1)^{1/p-1/q} \Vert f \Vert_q....
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Abel Means of an Integrable Function is Uniformly Convergent

I am studying Stein Shakarchi's book on Fourier Analysis. This page mentioned that the Abel means are absolutely convergent and uniformly convergent, just because $f$ is integrable. I did figure out ...
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54 views

Domain of a piecewise continuous function

In the picture above what is the domain of the piecewise-continous function? The first option is that the domain of the piecewise-continous function is [1,6] The second option is that the domain is $...
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Norm convergence of Fourier series in the dual of a Sobolev space

If $\mathbb{T}$ is the 1-torus and $1<p<\infty$, then for every $f$ in the Sobolev space $W^{1,p}(\mathbb{T})$ we have that the Fourier series of $f$ converges in the $W^{1,p}(\mathbb{T})$ to $f$...
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Fourier Analysis - Functions on a Circle

In general, if a continuous function $g(x)$ is defined on the interval $[-\pi,\pi]$, can I say it is definitely possible to extend this function to be a $2\pi$-period function? I think we need to make ...
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Wrong value of sum using fourier series

I have the following $2 \pi $ periodic function which is $t$ for $0\leq t <\pi$ and $0$ for $\pi\leq t<2\pi$ I'm asked to find its complex fourier series representation. So I calculate $c_k=\...
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Fourier cosine/sine series of $\cos x$

The question in my textbook asks to solve for the cosine and sine fourier series of $f(x) = \cos x$ on the interval $[0, \pi/2]$. This is my first PDE class. I tried integration by parts and got ...
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Fourier series calculate $f^2$

I have the function\fourier series $$ f(x) = \frac{1}{4} + \sum_{i=0}^\infty \frac{1}{3^k}\cos(2kx) $$ and I am asked to compute the integral $ \int_{0}^\pi f(x)^2 dx$ How do I proceed? I think ...
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Show that $\sum_{n\in\mathbb{Z}}|a_n|^2+|b_n|^2=4|c_0|^2+2\sum_{n\in\mathbb{Z}\setminus\{0\}} |c_n|^2$.

The $a_n,b_n$ and $c_n$ are Fourier coefficients. I start by expressing $a_n$ and $b_n$ in terms of $c_n$ as follows: since for every complex number $z$, $|z|^2=z\overline{z}$ we have that \begin{...
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How to express cosine of Fourier series as Fourier series again

I have the following Fourier series exapansion: \begin{equation} \phi(t) = a_0 + \Sigma_{n=1}^\infty (a_n\cos pnt + b_n\sin pnt). \end{equation} I want to express $\cos(\phi(t))$ as Fourier series ...