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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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Abel's test for uniform convergence in Fourier series

In Abel's test for uniform convergence, we wite the terms of the series as $u_n(x)=a_n f_n(x)$ and there is a condition which says that the functions $f_n(x)$ should be monotonic, in mathematical ...
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Proof that $\frac{t \xi_0}{\sqrt{\pi}} + \frac{1}{\sqrt{2\pi}} \sum \frac{\sin(nt)}{n}\xi_n$ is Wiener process.

Consider $\xi_i$ are i.r.v with standart normal distribution. Let $X_t = \frac{t}{\sqrt{\pi}}\xi_0 + \frac{1}{\sqrt{2\pi}} \sum_{n=1}^{\infty}\frac{\sin(nt)}{n}\xi_n$. We want to prove that $X_t$ is ...
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Intermediate Value Theorem applied to the Weierstrass function

Does anyone know how the intermediate value theorem applies to the Weierstrass function $$w(x)=\sum_{n\geq 0}a^n \cos(b^n\pi x),$$ $0 < a < 1 < b$, $ab > 1+3\pi/2$? More precisely, I ...
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How to comprehense the relation between convolution and each Fourier coefficient of f?

I am learning Fourier Analysis by Elias.M.Stein.I don’t understand the motivation of convolution.From my point of view,the convolutions corresponds to the “weighted averages” if we write it in ...
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Non-negative trigonometric polynomials to exponential form

I've been working on an exercise and have gotten stuck: Suppose that $T(x)=\sum_{n=0}^N a_n\cos(2\pi nx)+b_n\sin(2\pi nx)$ is non-negative on $[0,1]$. Show that there exist $c_0,...,c_N\in \mathbb{C}$...
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Compare R Programming with Sagemath for Fourier analysis? [on hold]

For Fourier mathematical simulation harmonics which one we will use R or Sagemath.
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Fourier series representation of piecewise function

$${Expand} \; f(x)= \begin{cases} 2{A\over L}x & 0\leq x\leq {L\over 2} \\ \\ 2{A\over L}\left(L-x\right) & {L\over 2}\leq x\leq L \end{cases} $$ I have determined $A_0$...
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How do i get fourier series of signals given below [on hold]

how can i calculate and plot $Ae^{-at}$ amplitude and phase spectrum? how can i determine the trigonometric fourier series of the signal given below ? Your help is appreciated !! Thanks
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Extending a diffeomorphism of $S^1$ to a diffeomorphism of $D^2$ with fourier series.

I'm reading this and there are some details that are missing. I'm asking for those. First let $f:S^1\to S^1$ a diffeomorphism. By Dini criterion we can write $f(e^{i\theta})=\sum_n \hat{f}(n)e^{in\...
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Fourier transform on $\mathbb{Z}_{2}^{d}$

Let $\mathbb{Z}_{2}^{d} = {\{\textbf{t} = (t_1, \ldots, t_d) : t_j \in \mathbb{Z}_2}\}$. Define the inner product on functions $f, g : \mathbb{Z}_{2}^{d} \rightarrow \mathbb{C}$ to be: $$\langle f, ...
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What does ordered pair notation mean in the context of Generalized Fourier Series? [on hold]

I remember my professor using this in lecture and understanding it at the time but I forgot and it came up on the last test and I didn't remember what it was, and I can't find it anywhere in my notes, ...
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Find $\sum_{n=0}^{\infty}{\frac{\cos(nx)}{n!}}$ [duplicate]

How could one find $$\sum_{n=0}^{\infty}{\frac{\cos(nx)}{n!}}\,?$$ I tried to use Fourier series and integrals depending on a parameter to reduce the problem to a differential equation, but that ...
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Fourier series Parseval equality with partial sums

Let $f \in \mathcal{R}_\left[-\pi,\pi\right]$ be function with period $2\pi$. We denote n-th partial Fourier series sum of function $f$ with $S_n(x)$. Prove that: $$ \int_{-\pi}^{\pi}\left(f(x)-S_n(...
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Derive fourier coefficient with IC dirac-delta

PDE: $$ u_t (x,t) - u_{xx} (x,t) = 0, \quad u(0,t)=0, \quad u_x(2,t)=0, \quad u(x,0)=\delta(x-x_0) $$ where $0<x_0<2$ is a given constant. I worked out my solution by using separation of ...
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Is there any result about this formula?

$${\sum_{k=0}^\infty \frac{\sin kx}{k}\over \sum_{k=0}^\infty \frac{\cos kx}{k}}$$ I want to use Euler formula but failed.I think it may be relevant to Fourier series so that you can transform the ...
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Evaluating sum $\sum_{m=0}^{\infty}\frac{(2-\delta_m^0)(-1)^m \lambda_0}{a(\lambda_0^2 -(\frac{m\pi}{a}))}\cos(m\pi x/a)$

How Can I evaluate the following sum$$\sum_{m=0}^{\infty}\frac{2-\delta_m^0}{a}\frac{(-1)^m \lambda_0}{\lambda_0^2 -(\frac{m\pi}{a})}\cos\left(\frac{m\pi x}{a}\right)=\frac{\cos(\lambda_0 x)}{\sin(\...
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Integral of $t^2e^{it^2}$

Is it allowed to solve this integral: $${I=\int_{-\infty}^{\infty}{t^2e}^{it^2}dt}$$ as follows? $$I=\lim_{a\rightarrow1}{\left[\int_{-\infty}^{\infty}{t^2e}^{iat^2}dt=\int_{-\infty}^{\infty}{-i\...
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Basis for subset of $\mathcal{L}^2(\mathbb{R})$ with the help of Fourier transform

Let $W \subset \mathcal{L}^2(\mathbb{R})$ be a linear subspace. I want to show that for a certain $\phi \in W$, $\{ \phi_m : m \in \mathbb{Z}\}$ is a basis for $W$. Here $\phi_m$ describes a ...
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Given a time-dependent function, complete it and find $c_0$

Complete graphically and analytically the function $f(t)$ so that the coefficients of the exponential Fourier series are pure imaginary: $$f(t)=\begin{cases}2t+1&\text{if $0\leq t\leq2$},\\\...
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Finding the sum of another Fourier series using Parseval's identity

So...I previously found the series for $ f(t)= \begin{cases} 0&\text{if}\, -\pi\leq t\lt -\pi/2\\ \cos(t)&\text{if}\, -\pi/2\leq t\leq \pi/2\\ 0&\text{if}\, \pi/2\lt t\leq \pi\\ \...
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How to find the conjugate of $\sum_{n=0}^{m}n \cos(2\pi x n)$?

Supposing I want to find the conjugate of $$\tag{1}\label{eq1}\sum_{n=0}^{m}n \cos(2\pi x n)$$ If I view (1) as a Fourier series then what would be the conjugate Fourier series and how would I go ...
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fourier series meaning complex radii meaning

I'm talking about the incredible answer from Mark Eichenlaub. Fourier transform for dummies He said : "we must allow the circles to have complex radii. It's the same thing as saying the ...
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Having trouble with undefined terms in a Fourier series

So...I am trying to find the Fourier series for the following function: $ f(t)= \begin{cases} 0&\text{if}\, -\pi\leq t\lt -\pi/2\\ \cos(t)&\text{if}\, -\pi/2\leq t\leq \pi/2\\ 0&\...
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Unexpected results for Fourier synthesis using IFFT in 2D

I am trying to recover a 2D function using inverse DFT, to my understanding the IDFT outputs the coefficients of the fourier series of the original function up to the Nyquist frequency. So for ...
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Fourier Sine Series Question with modulus function

How do I go about calculating the fourier series of: $$x(π-|x|)$$ $$over $$$$[-π,π]$$ I notice that it is an odd function, therefore a0 and an equal 0, therefore we only have to find bn. However I ...
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Showing $\int_{-\infty}^{\infty}\frac{\sin^2(\omega)}{\omega^2}d\omega=\pi$

Given the function $f$ with $f(t)=1$ for $|t|<1$ and $f(t)=0$ otherwise, I have to calculate its Fourier-transform, the convolution of $f$ with itself and from that I have to show that $$\int_{-\...
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PDE, Dirichlet problem for a circle

When I'm learning laplace's equation with the case is dirichlet problem for a circle, i know that naturally i have to separate variables in polar coordinate. And i derive from rectangle to polar ...
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PDE-solution with Fourier series

Consider the problem where PDE given with boundary and initial conditions: $u_{tt}+2\alpha u_t-u_{xx} = 0, \ 0<x<\pi, \ t>0 \\ u(x,0)=f(x), \ u_t(x,0)=0,\ 0<x<\pi \\ u(0,t)=u(\pi,t)=0 ...
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Prove that a function is equal to its Fourier Series

Find the Fourier Series of the function $p(x)=\frac{\cos(5\pi x)+e^{-i\pi x}+1}{2}$ and justify why $p(x)$ must equal its Fourier series. This is a review problem for my Analysis exam. My idea was ...
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How to find the coefficient of heat equation solution with non-odd initial condition

I've got a situation where I'm supposed to solve the 1-D heat equation $v_t = c^2 u_{xx}$ under boundary conditions and an initial condition, and so after some run-of-the-mill process, I attained the ...
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Calculate the given sum

Problem: $$\sum_{k=1}^{\infty} \frac{1}{k^2(k+1)^2}$$ This is a problem from a course of Fourier Series and the only hint was to use partial fraction decomposition. I'm sure that this is meant to be ...
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Fourier series for a periodic vector field

In usual Fourier analysis, we have a periodic function like $f: \mathbb{R}^3 \rightarrow \mathbb{C}$. The function can be decomposed into a countable basis (i.e. $\{\mathrm{e}^{i\mathbf{k_n}\cdot \...
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Find fourier series of piecewise function

Find the Fourier series of $f :(-\pi, \pi) \rightarrow \mathbb{R}, x \mapsto f(x) = \begin{cases} \frac{-\pi}{2} & -\pi < x < 0 \\ \frac{\pi}{2} & 0 < x < \pi . \...
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How to compute the sum of series using the Fourier series expansion of the function $f(x) = x^2$ over $(0,2\pi].$

Consider the $2\pi$ periodic function $f(x) = x^2$ defined over the interval $x\in (0,2\pi].$ The Fourier expansion of $f$ is as follows: $$f(x) = \frac{4\pi^2}{3}+\sum_{n\geq 1}\left(\frac{4}{n^2}\...
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Solving the biharmonic equation using Fourier series

I'm trying to solve the biharmonic equation $\nabla^4 \phi = 0$ for $\phi(x,y)$ over a square domain $0 \le x \le L_x$, $0 \le y \le L_y$, for (currently) general boundary conditions. However, I end ...
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discontinuity of $f(x) = \sum_{n\geq 1} \sin((n+ \alpha /n)x)/n$

It is well known that the function defined by $$g(x) = \sum_{n\geq 1 } \sin(nx )/n $$ is a piece-wise linear function $x$, and has jumps at $x = 2 m \pi $. Its picture is Now let us consider a ...
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Integrate Fourier Series and find sum

By integrating the Fourier series equation $$y(x) = \frac{\pi}{2} - \frac{4}{\pi} \sum_{m=0}^{\infty} \frac{cos((2m+1)x)}{(2m+1)^2} $$ term by term from 0 to x, find the function $g(x)$ whose Fourier ...
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Solve a PDE with plane polar coordinates (method of separation)

Solve $∇^2(u) = 0$ in two dimensions for $r < 1$ (in plane polar coordinates), with boundary conditions $u(1, θ) = A$ for $0 < θ < π$ , $0$ for $π < θ < 2π$ where $A$ is a given ...
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Bessel coefficients decay

Is it possible to bound the decay of the coefficients of the bessel function $J_n(j^n_lr)$, where $j_l^n$ is the n-th zero of $J_n$? I want to know if there is a bound in terms of $n$ and $l$
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A lemma in the proof of Fourier's Main Theorem.

Assume $f,g:[-\pi,\pi]\to\mathbb{R}$ are continuous functions. We define the $L^2$ norm of $f$, and the scalar product between $f$ and $g$ as $$\|f\|_{L^2}=\sqrt{\int_{-\pi}^{\pi}|f(x)|^2\,dx}$$ and $$...
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$f$ is a $2\pi$ periodic continuously differentiable function on $[-\pi,\pi]$

The question : Let $f$ be a $2\pi$ in $C^1$ function on $[-\pi,\pi].$ Let $c_n$ be the Fourier coefficient of $f: c_n := \hat{f}(n)= \frac{1}{2\pi} \int_{-\pi}^{\pi}f(x)e^{inx}dx$. Show that $\sum_{...
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Phase angle of a Fourier series

I have read in my textbook that if a Fourier series consists of only sine terms(that is, the function is odd), its phase angle is 0. If the Fourier series consists of only cosine terms(that is, the ...
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If $f$ is continuous and $\hat{f}$ is supported on $S$ then its Fourier series converges

In general $\scriptstyle\text{(even if it is hard to show a simple counterexample)}$ that $f$ is continuous doesn't mean its Fourier series converges everywhere. In this question it is shown that if $...
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How to Sketch a graph of a extended function from one intervel to another intervel

I have already done part a,b,c. But in d part they ask to extend the function. I can't understand how to extend it and sketch the extended function.
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Find the fourier transform of $e^{-ax^2}\cos(bx)$

I am trying to find the Fourier transform of the function $e^{-ax^2}\cos(bx)$. I am changing $\cos(bx)$ to its exponential form but after that I am getting stuck: can you guide me showing how to find ...
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1answer
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Reading Fourier Transform Tables - Are They Symmetric?

We use the definitions $$F(k)=\mathcal{F}(f(x))=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} e^{-ikx}f(x) \ dx,$$ where the inverse is defined as $$f(x)=\mathcal{F}^{-1}(F(k))=\frac{1}{\sqrt{2\pi}}\...
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How to prove: $f \in C^1([0, \pi])$ with $f(0)= f(\pi) = 0$, then $\int_0^{\pi} |f|^2 dx \le \int_0^{\pi} |f'|^2 dx$. [duplicate]

Problem Prove that if $f \in C^1([0, \pi])$ with $f(0)=f(\pi) = 0$, then $$ \int_0^\pi |f|^2 dx \le \int_0^\pi |f'|^2 dx.$$ I want to prove it using Fourier sine series of $f$. Let $f \sim \sum_{n=...
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Fourier Series of exp(x) and its integral

I have to find the Fourier-coefficients of $f(x)=\text{exp}(x)$ for $-1 < x <1$ and evaluate the value of this series at $x=2$. (EDIT: with period 2) I have calculated the coefficients to $$...
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Fourier Series $f(x)=-1$ for $-\pi<x<0$, $f(x)=x$ for $0<x<\pi$ and expansion help

Apologies in advance as I'm sure you get many similar questions. So I have to expand the Fourier Series $f(x)=-1$ for $-\pi<x<0$, $f(x)=x$ for $0<x<\pi$ I end up with $\frac {a_0}{2}$(ie ...
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1answer
47 views

Finding Fourier series of cosh

I have to find an even Fourier series (i.e. only cosine-terms) of $$f(x)=\cosh(x-1)$$ in the interval $0 \leq x \leq 1$ with periodicity 2. (EDIT: With my calculated Fourier coefficients I end up ...