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.
3,850 questions
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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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0answers
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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.
2
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1answer
22 views
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 $A_0$...
-1
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0answers
25 views
How do i get fourier series of signals given below
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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1answer
43 views
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\...
2
votes
1answer
30 views
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: $$\langle f, ...
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0answers
13 views
What does ordered pair notation mean in the context of Generalized Fourier Series? [on hold]
I remember my professor using this in lecture and understanding it at the time but I forgot and it came up on the last test and I didn't remember what it was, and I can't find it anywhere in my notes, ...
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2answers
51 views
Find $\sum_{n=0}^{\infty}{\frac{\cos(nx)}{n!}}$ [duplicate]
How could one find $$\sum_{n=0}^{\infty}{\frac{\cos(nx)}{n!}}\,?$$
I tried to use Fourier series and integrals depending on a parameter to reduce the problem to a differential equation, but that ...
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0answers
24 views
Fourier series Parseval equality with partial sums
Let $f \in \mathcal{R}_\left[-\pi,\pi\right]$ be function with period $2\pi$. We denote n-th partial Fourier series sum of function $f$ with $S_n(x)$. Prove that:
$$
\int_{-\pi}^{\pi}\left(f(x)-S_n(...
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1answer
18 views
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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2answers
44 views
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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2answers
156 views
+50
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$$\sum_{m=0}^{\infty}\frac{2-\delta_m^0}{a}\frac{(-1)^m \lambda_0}{\lambda_0^2 -(\frac{m\pi}{a})}\cos\left(\frac{m\pi x}{a}\right)=\frac{\cos(\lambda_0 x)}{\sin(\...
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1answer
87 views
Integral of $t^2e^{it^2}$
Is it allowed to solve this integral:
$${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\...
1
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1answer
17 views
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 ...
1
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1answer
18 views
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: $$f(t)=\begin{cases}2t+1&\text{if $0\leq t\leq2$},\\\...
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1answer
53 views
Finding the sum of another Fourier series using Parseval's identity
So...I previously found the series for
$
f(t)=
\begin{cases}
0&\text{if}\, -\pi\leq t\lt -\pi/2\\
\cos(t)&\text{if}\, -\pi/2\leq t\leq \pi/2\\
0&\text{if}\, \pi/2\lt t\leq \pi\\
\...
1
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1answer
75 views
How to find the conjugate of $\sum_{n=0}^{m}n \cos(2\pi x n)$?
Supposing I want to find the conjugate of
$$\tag{1}\label{eq1}\sum_{n=0}^{m}n \cos(2\pi x n)$$
If I view (1) as a Fourier series then what would be the conjugate Fourier series and how would I go ...
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1answer
25 views
fourier series meaning complex radii meaning
I'm talking about the incredible answer from Mark Eichenlaub.
Fourier transform for dummies
He said :
"we must allow the circles to have complex radii. It's the same thing
as saying the ...
0
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2answers
32 views
Having trouble with undefined terms in a Fourier series
So...I am trying to find the Fourier series for the following function:
$
f(t)=
\begin{cases}
0&\text{if}\, -\pi\leq t\lt -\pi/2\\
\cos(t)&\text{if}\, -\pi/2\leq t\leq \pi/2\\
0&\...
0
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0answers
10 views
Unexpected results for Fourier synthesis using IFFT in 2D
I am trying to recover a 2D function using inverse DFT, to my understanding the IDFT outputs the coefficients of the fourier series of the original function up to the Nyquist frequency.
So for ...
0
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0answers
15 views
Fourier Sine Series Question with modulus function
How do I go about calculating the fourier series of:
$$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 ...
1
vote
2answers
67 views
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
$$\int_{-\...
0
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1answer
36 views
PDE, Dirichlet problem for a circle
When I'm learning laplace's equation with the case is dirichlet problem for a circle, i know that naturally i have to separate variables in polar coordinate.
And i derive from rectangle to polar ...
0
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0answers
18 views
PDE-solution with Fourier series
Consider the problem where PDE given with boundary and initial conditions:
$u_{tt}+2\alpha u_t-u_{xx} = 0, \ 0<x<\pi, \ t>0 \\ u(x,0)=f(x), \ u_t(x,0)=0,\ 0<x<\pi \\ u(0,t)=u(\pi,t)=0 ...
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0answers
30 views
Prove that a function is equal to its Fourier Series
Find the Fourier Series of the function $p(x)=\frac{\cos(5\pi x)+e^{-i\pi x}+1}{2}$ and justify why $p(x)$ must equal its Fourier series.
This is a review problem for my Analysis exam.
My idea was ...
1
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1answer
28 views
How to find the coefficient of heat equation solution with non-odd initial condition
I've got a situation where I'm supposed to solve the 1-D heat equation $v_t = c^2 u_{xx}$ under boundary conditions and an initial condition, and so after some run-of-the-mill process, I attained the ...
0
votes
2answers
69 views
Calculate the given sum
Problem: $$\sum_{k=1}^{\infty} \frac{1}{k^2(k+1)^2}$$
This is a problem from a course of Fourier Series and the only hint was to use partial fraction decomposition.
I'm sure that this is meant to be ...
0
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0answers
23 views
Fourier series for a periodic vector field
In usual Fourier analysis, we have a periodic function like $f: \mathbb{R}^3 \rightarrow \mathbb{C}$. The function can be decomposed into a countable basis (i.e. $\{\mathrm{e}^{i\mathbf{k_n}\cdot \...
0
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0answers
22 views
Find fourier series of piecewise function
Find the Fourier series of $f :(-\pi, \pi) \rightarrow \mathbb{R}, x \mapsto
f(x) = \begin{cases}
\frac{-\pi}{2} & -\pi < x < 0 \\
\frac{\pi}{2} & 0 < x < \pi .
\...
2
votes
1answer
29 views
How to compute the sum of series using the Fourier series expansion of the function $f(x) = x^2$ over $(0,2\pi].$
Consider the $2\pi$ periodic function $f(x) = x^2$ defined over the interval $x\in (0,2\pi].$
The Fourier expansion of $f$ is as follows:
$$f(x) = \frac{4\pi^2}{3}+\sum_{n\geq 1}\left(\frac{4}{n^2}\...
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0answers
12 views
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 ...
1
vote
1answer
22 views
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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0answers
18 views
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 ...
1
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1answer
17 views
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 ...
0
votes
0answers
8 views
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 $l$
1
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1answer
41 views
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 $$...
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1answer
52 views
$f$ is a $2\pi$ periodic continuously differentiable function on $[-\pi,\pi]$
The question :
Let $f$ be a $2\pi$ in $C^1$ function on $[-\pi,\pi].$ Let $c_n$ be the Fourier coefficient of $f: c_n := \hat{f}(n)= \frac{1}{2\pi} \int_{-\pi}^{\pi}f(x)e^{inx}dx$. Show that $\sum_{...
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0answers
12 views
Phase angle of a Fourier series
I have read in my textbook that if a Fourier series consists of only sine terms(that is, the function is odd), its phase angle is 0.
If the Fourier series consists of only cosine terms(that is, the ...
0
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0answers
18 views
If $f$ is continuous and $\hat{f}$ is supported on $S$ then its Fourier series converges
In general $\scriptstyle\text{(even if it is hard to show a simple counterexample)}$ that $f$ is continuous doesn't mean its Fourier series converges everywhere.
In this question it is shown that if $...
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0answers
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How to Sketch a graph of a extended function from one intervel to another intervel
I have already done part a,b,c. But in d part they ask to extend the function. I can't understand how to extend it and sketch the extended function.
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0answers
41 views
Find the fourier transform of $e^{-ax^2}\cos(bx)$
I am trying to find the Fourier transform of the function $e^{-ax^2}\cos(bx)$.
I am changing $\cos(bx)$ to its exponential form but after that I am getting stuck: can you guide me showing how to find ...
0
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1answer
32 views
Reading Fourier Transform Tables - Are They Symmetric?
We use the definitions $$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}}\...
1
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0answers
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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=...
2
votes
1answer
42 views
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
$$...
0
votes
0answers
28 views
Fourier Series $f(x)=-1$ for $-\pi<x<0$, $f(x)=x$ for $0<x<\pi$ and expansion help
Apologies in advance as I'm sure you get many similar questions.
So I have to expand the Fourier Series $f(x)=-1$ for $-\pi<x<0$, $f(x)=x$ for $0<x<\pi$
I end up with $\frac {a_0}{2}$(ie ...
0
votes
1answer
47 views
Finding Fourier series of cosh
I have to find an even Fourier series (i.e. only cosine-terms) of $$f(x)=\cosh(x-1)$$ in the interval $0 \leq x \leq 1$ with periodicity 2.
(EDIT: With my calculated Fourier coefficients I end up ...
1
vote
0answers
17 views
On a simple discrete Fourier transform
Given an integer N, and two non-negative integer valued variables x,y which take values in ${0,1,...,N-1}$. Is it possible to obtain a close form for the following summation?
$$f(x,y)=\frac{1}{N}\...
1
vote
1answer
29 views
A consequence of Riemann-Lebesgue lemma
Riemann Lebesgue lemma states that: For a function $f\in L_1([-\pi,\pi])$ we have
\begin{align}
\lim_{|n|\to \infty} \int_{[-\pi,\pi]} f(t) e^{-int} \text{ d}t =0.
\end{align}
The integral here is ...
0
votes
0answers
17 views
singularity of $\sum_{n\in \mathbb{Z}} e^{in x }/ (n+\alpha)^2 $
Here $\alpha $ is a non-integral real number.
It is apparent that the series converges uniformly. So it converges to a continuous function.
The point is that, as the coefficients decay just like $...
0
votes
2answers
28 views
Proving parseval identity for trigonometric polynomials
Show that
$$||P||_2^2=\sum_{k=-N}^N \langle P,e_k\rangle^2,$$
where $e_k$ are the Fourier basic functions, and $P$ is a trigonometric polynomial of degree $N.$
I am not sure how to link trigonometric ...
