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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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Minimizing the MSE of a Fourier series

When approximating an odd function with period $2\pi$ by a Fourier-Sine-Series with $m$ terms, it has the error $$E_m=\int_{-\pi}^\pi \left[f(x)-\sum_{n=1}^m b_n \sin(nx)\right]^2 dx$$ Now I have to ...
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Is it possible to use FFT to derive a Fourier series fitting to data?

I want to do something like what is done in this question about fitting , ie find a Fourier series that approximates a continuous but complicated function. However I want to know whether it is ...
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2answers
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How do I periodically extend a function?

I am given that $f(x) = -1-x$ on $[-1,1)$ and I'm asked to find the Fourier series for the $2$ period extension of $f(x)$. I get how to find a Fourier series for the given function on the given ...
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Periodic extension of real valued function and its Fourier series [on hold]

Consider the periodic extension of the real valued function $$ f(t) =(t-1/2)^2, t \in(0,1) $$ and construct its fourier series. After, calculate the sum $$\sum_{n=0}^n 1/n^2 $$
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Integrals of products of sines and cosines with arbitrary periods

I am currently studying the Fourier series, which involves integrals of products of sine and cosine functions. Because sine and cosine are orthogonal, we have been using the following facts to help us ...
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30 views

Fourier Series - Integration Limits help

So I am doing separation of variables and I reach the stage $$ \sum_{n=1}^{\infty} Q_n \cos(P_nx) = 2 $$ where $$f(x) = 2. $$ Now this is similar to the Fourier series where the $Q_n$s are the ...
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Triple sum $\sum\limits_{a=1}^{\infty} \sum\limits_{b=1}^{\infty} \sum\limits_{c=1}^{\infty} \frac{\cos a \cos b \cos c}{a^2 + b^2 + c^2}$

We have poor water heating system in our countryside house (currently it takes 4 hours to warm up the water), and my father has decided to improve it; he bought a water tank and placed it up in the ...
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Is it possible that $\sin^5 (x)$ doesn't have a Fourier series? [on hold]

As a math project I have to find the Fourier series of this function over $[-\pi,\pi]$ and I have tried integration for the coefficients and also complex numbers with the binomial theorem and what ...
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Finding the value of a Fourier series

Expand the given functions in an appropriate cosine or sine series. $$ f(x) = \begin{cases} 1, & -2 < x < -1 \\ -x, & -1 \leq x \leq 0 \\ x, & 0 \leq x \leq 1 \\ 1, &...
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1answer
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Strategies for proving continuity and differentiability of trigonometric series

Let $f$ be a function defined by a series $$f\left(x\right)=\sum c_n e^{inx}.$$ Sometimes, I can prove that the series converges pointwise (when it does), using the Dirichlet test. When the ...
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Fourier series confusion

Find the Fourier series of Question 1 $$ f(x) =\begin{cases} 0&&\text{for $-1 < x < 0$}\\\\ x&&\text{for $0 \leq x \leq 1$} \end{cases} $$ and Question 2 $$f(x) = x + \pi,\;...
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1answer
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Fourier series matlab

I have examples of Fourier series and I would using the Matlab I know how calculate the coefficients of Fourier series in Matlab but how calculate the summation ? do I use “For loop “ or what in ...
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24 views

maximum of the function $f(x) = \sum_{n\geq 1 } \sin n x /n^\alpha $

Here $\alpha >1 $. The function is defined as $$f(x) = \sum_{n\geq 1 } \sin n x /n^\alpha .$$ The domain is $(0, \pi)$. We know that if $\alpha = 1$, $f(x) = (\pi -x )/2$. If $\alpha >1$, ...
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Help with Incorrect proof of Plancherel's Theorem.

Plancherel's Theorem Let $f$ $\in$ $(C^{2\pi}$, $||f||_{2}$) where $||f||_{2} = \Big(\frac{1}{2\pi}\int_{-\pi}^{\pi}|f(x)|^{2}dx\Big)^{\frac{1}{2}}$. \begin{align*} ||f-S_{n}(f)||_{2} \overset{...
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1answer
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Fourier series of translated square

I can't seem to find the correct Fourier series coefficients ($s_n$) of the following periodic function. I know how to get the Fourier series of the same one that is not vertically translated and has ...
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ODE with Fourier Series as forcing function . How to get a1? i have found ao and b1 but cant find a1 at all

How do I find $a_{1}$? I have found how to find $a_{0}$ and $b_{1}$, but I can't seem to find $a_{1}$ at all. Please advise.
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72 views

How do I evaluate this integral

$$\frac{2}{\pi}\int_{0}^{\pi}e^{3x}\sin nx\,dx$$ I know I have to use part integration and I get here $$\frac{2}{3}\pi*\sin nx*e^{3x}-\int_{0}^{\pi}e^{3x}(\sin nx)'\,dx$$ I don't know how to put ...
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2answers
74 views

Asymptotics of $\sum_{n\geq 1 } \frac{\sin n^2 t }{ n^2 } $

This is the Riemann function. I would like to determine its asymptotics as $t \rightarrow 0^+ $. First let $t=x^2 $, so that we treat the series $$ \sum_{n\geq 1 } \frac{\sin n^2 x^2}{ n^2}. ...
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Use an inner product to find the projection onto the span of Fourier approximation

Using the inner product on $C[-\pi,\pi]$ given by $$ \langle f,g\rangle = \int_0^1 f(t)g(t)dt $$ Find the projection of $h(x) = x$ onto the span of $$\{1, \sin x, \cos x, \sin(2x), \cos(2x),\ldots,\...
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How to decide on which $\ n$-th term should I start the sum of a Fourier series

I have recently started to study Fourier series and I have the following function: $ f(x)= \begin{cases} 1&\text{if}\,& 0\leq x\leq \pi \\ -1&\text{if}\,& \pi<x\leq 2\pi\\ \end{...
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2answers
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How to calculate the Fourier Coefficient of $\sin^5(x)$ over $[-\pi,\pi]$? [closed]

I would have to integrate $\sin^5(x)\cdot\sin(nx)$, but I have no idea how to. And that's the only coefficient I need for the series.
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How to solve given integral? [closed]

I need it for my Fouier series's coefficent. $$ \int_{-π}^{π}\left| x\right| \cos{5x} \, dx $$
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How to find factor (coefficient) in given function Fourier series?

$$f(x)=x^3+\sin x$$ is given function. Interval is: $$(-π,π)$$ Fourier series is $$f(x)=\frac{a_{0}}{2}+\sum \limits_{n=1}^{\infty}(a_{n}\cos(nx)+b_{n}\sin(nx))$$ I have to find $$b_{5}$$
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Asymptotic behavior of $f(x) = \sum_{n\geq 1 } a_n \cos n x $

Here the coefficients $a_n$ are positive and decreasing in a neighborhood of $+\infty$. Moreover, there is the limit $$\lim_{n\rightarrow\infty } \frac{a_n }{n^{-3/2}}= \alpha > 0 .$$ What is ...
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52 views

Differentiability of the function defined by $\sum_{n\geq 1 } \sin n^2 x / n^{5/2} $

It is known that the Riemann function $$ \sum_{n\geq 1 } \frac{\sin n^2 x }{n^2} $$ is differentiable only at countably many points. How about the functions $$\sum_{n\geq 1 } \frac{\sin n^2 x}{...
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Asymptotics of the Weierstrass function

It is known that the Weierstrass function $$f (x) =\sum_{n\geq 1 } a^n \cos (b^n x ) , $$ with $0 < a < 1$, $ab>1$, is nowhere differentiable. But this should not prevent us from studying ...
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Series expansion of $\ln(1+\exp(ix))$

I am looking for a way to prove the following identity $$\ln(1+\exp(ix))= \sum_{n=1}^{\infty}\frac{(-1)^{n+1} \exp(inx)}{n} $$ What I know about the function is the given function is periodic with ...
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If $f \in L^{1}(\mathbb{T})$ and $\sum_{n \in \mathbb{Z}} |{\hat{f}(n)}| < \infty $ then $f \in C^{0}(\mathbb{T})$

Where $ \mathbb{T} = \mathbb{R}/2\pi\mathbb{Z}.$ My initial thoughts are that if I can show that $ \frac{1}{2\pi}(f*F_{N}) \rightarrow f$ uniformly then I can use a previous result to show that $f \...
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Dirichlet conditions for the existence of the Fourier transform

Assuming a periodic function, Dirichlet conditions are sufficient (not necessary) conditions for Fourier series. 1) As they are defined for Fourier series, how Dirichlet conditions can also be ...
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odd extension function and Dirichlet problem

so I need to find the odd extension function for : $\phi (x) = \begin{cases}x& \text{ , } 0 \leq x\leq 1 \\ 1& \text{ , } 1 \leq x\leq 2\\ 3-x& \text{ , } 2\leq x\leq 3 \end{cases}$...
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Proving that $\sum_{n=1}^\infty \frac{\sin^2 n}{n^2}=\sum_{n=1}^\infty \frac{\sin n}{n}$.

Proving that $$\sum_{n=1}^\infty \frac{\sin^2 n}{n^2}=\frac{\pi -1}{2}$$ I've known a similar conclusion $$ \sum_{n=1}^\infty \frac{\sin nx}{n}= \begin{cases} \dfrac{\pi - x}{2} & x \in (0, 2\pi),...
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Playing with the Definition of Fourier Series Again, This Time with Even trick

$\textbf{The Problem:}$ Suppose that for any smooth function $F:[-L,L]\to\mathbb C$ satisfying $F(L)=F(-L)$ we can write for some $c_n\in\mathbb C$ $$F(x)=\sum^{\infty}_{n=-\infty}c_ne^{in\pi x/L}.$$...
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1answer
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Manipulating Definition of Fourier Series

$\textbf{The Problem:}$ Suppose that for any smooth function $F:[-L,L]\to\mathbb C$ satisfying $F(L)=F(-L)$ we can write for some $c_n\in\mathbb C$ $$F(x)=\sum^{\infty}_{n=-\infty}c_ne^{in\pi x/L}.$$...
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Uniqueness of solutions when input is discontinuous at $t_0$.

I have recently been reading Signal Processing and Linear Systems by B. P. Lathi, but I also have some general knowledge of differential equations from reading parts of Elementary Differential ...
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28 views

Fourier series complex functions $f : \mathbb C \rightarrow \mathbb C$

I'm starting to learn about Fourier series. For now it is really formal and I don't see yet the boundaries of what I'm studying. I was nevertheless curious to know if it makes sense to talk about ...
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23 views

What does the plot of the Fourier series of f look like?

Consider the Fourier series of f, defined using the interval −1 ≤ x ≤ 1. Plot f together with its Fourier series (the plot should show several periods of the Fourier series). (i) $$f(x) = 1$$ (ii) $$f(...
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How to solve a Semi-infinite Fourier Transform of the Laplace Equation with Dirac Delta boundaryCondition

I am given:enter image description here I know that I will need to use the Semi-infinite Sine transform to solve this problem, however I am unsure how to go about solving with a Dirac Delta function ...
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25 views

Discrete Fourier transform terms derivation

According to the paper Lecture 7 - Discrete Fourier Transform we can approximate a Fourier transform $$F(\omega ) = \int_{ - \infty }^\infty {f(t){e^{ - j\omega t}}}$$ by the series $$F(\omega ) = \...
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Fourier coefficients double half-wave rectifier

I’d like to find the Fourier coefficients of $$ x(t)=|A \cos(2 \pi f_0 t )| $$ The period is $ T_0 / 2 $ so i applied the definition $$\frac{1}{T}\int_{0}^{L} |A \cos(2\pi f_0 t )| \cos \left(\frac{k ...
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1answer
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Solution to Diffusion Equation in similar (but slightly different) situations

On an old PDE exam I'm looking through for my finals, a multiple-part question comes as: (a) Solve the diffusion equation with mixed Neumann-Dirichlet BC's: \begin{align} &u_t=u_{xx}, \quad 0&...
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1answer
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Equivalent solutions to Fourier series of $e^x$

I wanted to find the Fourier series of $e^x$ on $[-L,L]$. Using the Complex Form of $$f(x) = \sum_{n=-\infty}^{\infty} C_n e^{i \frac{n\pi}{L}x}$$ with $$C_n = \frac{1}{2L} \int_{-L}^L f(x) e^{i \...
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Confusion regarding Fourier half series for sine and cosine

I have been struggling with a problem for a long time. Solving a second order partial differential equation using Fourier half series in sine with the help of Mathematica gives me $$ \phi_{mine}(x,y)=-...
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1answer
22 views

Help me understand how to use a Fourier Series to calculate an Σ sum

So, we're given a function $f(x) = \begin{cases} 2, &-\pi < x\le 0 \\ 6, &0 < x\le\pi \end{cases}$, while $f(x+2π) = f(x)$ for any $x\in\Bbb R$. Now, I've calculated the Fourier series ...
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Fourier coefficients of $f\left(z\right)=\frac{1}{1+\cos z}$ through Laurent series

I had to find the Fourier coefficients of this simply periodic function $$f\left(z\right)=\frac{1}{1+\cos (z)},$$ I proceeded considering the $w=exp(iz)$ and considering the Laurent expansion of the ...
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The number $\pi$ in an unexpected context

[This is a follow-up question to this one: Figures and Numbers: Relating properties of geometric shapes and their Fourier series.] Drawing shapes by some predefined Fourier series I found this square ...
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1answer
30 views

Fourier series of $\cos^4(x)$

Expand $\cos^4(x)$ into a Fourier series. we already know that we need to find $\int_{-\pi}^\pi f(x)dx$ which will = $\frac{3\pi}{4}$. now we need to find $a_n = \int_{-\pi}^\pi \cos^4(x) \cos(nx) ...
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180 views

Figures and Numbers: Relating properties of geometric shapes and their Fourier series

Consider two types of parametrized curves $\gamma:[0,2\pi]\rightarrow \mathbb{R}^2$ open curves $\gamma_\sim(t) = (t,a(t) + b(t))$ closed curves $\gamma_\bigcirc(t) = (a(t),b(t)) = a(t) + ib(t)$ ...
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1answer
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Fourier series convergence 1

For a square integrable function $f$ on $[-\pi ,\pi]$ fourier series of $f$ converges to $f$ in the sense $\lim\limits_{n \rightarrow \infty} \| f - S_n \| = 0$ where $$S_n(t) = \frac {a_0} {2} + \sum\...
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1answer
71 views

Fourier series of regular polygons

The definition of a regular polygon by two real-valued functions $(x(t)$, $y(t))$ – or alternatively by a complex-valued function $x(t) + iy(t)$ – suggests to calculate the Fourier series $...
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2answers
85 views

Fourier series of some sawtooth functions

Given a periodic function $f:[0,2\pi] \rightarrow \mathbb{R}$ one calculates the Fourier coefficients $a_k$ and $b_k$ by $$a_k \sim \int_0^{2\pi}f(t)\cos(kt)\mathrm{d}t$$ $$b_k \sim \int_0^{2\pi}f(t)...