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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Coefficient Calculation on Fourier Series Under two minutes, Yes, How?!

Example of one Question for preparing the entrance exam: Fourier series of function: $$ f(x)=f(x+2\pi), f(x) =\left\{ \begin{array}{rcr} 1 & & -\pi <x<0 \\ \sin x & &...
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Question about function with Fourier series and Hölder continuity at a point on the circle

Let $(\lambda_{n})$ be lacunary (i.e. $\exists$ constant $q>1$ such that $\lambda_{n+1}>q\lambda_{n}$ for all $n\in\mathbb{N}$); $f\in L^{1}(T)$ with Fourier series $\sum_{n\in\mathbb{N}}a_{n}\...
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28 views

The set of all maximal ideals (Wiener Algebra)

I am trying to prove a proposition and in my proof I somehow need to find the set of all maximal ideals of a Banach Algebra. This is my working environment: Let $A(\mathbb{R}^2)$ be the (Wiener ...
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320 views

Complex form of Fourier Series

So, the last part of the university syllabus in the chapter of Fourier Series is: ...
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3answers
319 views

Origin of coefficients of fourier series?

I was wondering how we derive these formulas, and why we have a separate formula for $a_0$? All I know from advanced engineering mathematics text book are following formulas but where do they come ...
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Compute Fourier coefficients of spline fit to data

Suppose you have data $$\{(x_i, 1)\}_{1\dots N}, \quad x_1=0<x_i<x_{i+1}<x_N=2\pi \ \forall i \in\{2,\dots N-1\} $$ In other words, we have a sequence of $y=1$ values at distinct random ...
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2answers
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Solving non-homogenous PDE with forcing function (which diappears!) dependent only on time

Applying the method of eigenfunction expansion to the PDE $$u_t -c^2u_{xx}=F(t)$$ $$0<x<L, t>0$$ $$u(x,0)=f(x)$$ $$u_x(0,t)u_x(L,t)=0$$ for the homogenous part of this equation ($L[v(x,t)]=0$...
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Resonance in wave equation

I have solved the non-homogenous equation by the method of eigenfunction expansion $$u_{tt} - c^2 u_{xx}=F(x)\sin(\omega t)$$ $$0<x<L, t>0$$ $$u(x,0)=u_t(x,0)=0$$ $$u(0,t)=u(L,t)=0$$ and got ...
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Absolute convergence and continuity imply uniform convergence of Fourier series to the function?

I encounter the following problem: Let $f$ be a periodic continuous function in $[0,2\pi]$ such that the Fourier of $f$ is absolute convergence, that is $$|a_0|+\sum_{n=1}^\infty (|a_n|+|b_n|)<\...
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2answers
136 views

Calculate $\sum\limits_{n=1}^{\infty} \frac{a\cos(nx)}{a^2+n^2}$

I have to calculate $\sum\limits_{n=1}^{\infty} \frac{a\cos(nx)}{a^2+n^2}$ for $x\in(0,2\pi)$. I have used the function $f(x)=e^{ax}$ and I have calculated the Fourier coefficients which are: $$a_0=\...
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Getting the DFT of irregularly spaced points

I am trying to estimate the discrete Fourier transform of a discrete surface, $x:\{1,\dots,N\}\times \{1,\dots,N\} \to\mathbf{R}$, given a sparse set of samples on the grid. If we had all the ...
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1answer
587 views

Fourier Series Reduced Form: Phase Angle and Spectra

Im very confused regarding how to determine the angle on the reduced or harmonic form representation of the Fourier series. Some books state the following: $$f(t)=F_0+\sum_{n=1}^\infty |F_n |\cos(n\...
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29 views

The complex version of the Riemann-Lebesgue lemma

I can't prove the complex version of the Riemann-Lebesgue lemma. $$ f(x) \in \mathbf{C} \\ a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos nx \, dx, \quad b_n = \frac{1}{\pi} \int_{-\pi}^{\...
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1answer
28 views

Geometrical interpretation of complex exponential integral

Coefficients of Fourier series of a function $f$ are computed by multiplying $f(x)$ by the exponential term $e^{-inx}$, then by integrating $f(x)e^{-inx}$ from $-\pi$ to $\pi$ and dividing by $2\pi$ (...
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1answer
104 views

A Continuous Function with a Divergent Fourier Series

This is a Q&A; I hope simply posting a question and then answering it is the right protocol. This is stuff I thought everybody knew, but in at least two recent threads it's turned out to be somewhat ...
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65 views

Confused computing sum of Fourier series

I am having some issues understanding Fourier series and I am stuck trying to solve a problem. So the function $u$ has period $2\pi$ and is defined as $$u(x) = \begin{cases} 1 & 0 \leq x \...
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18 views

Original statement of Wiener's $1/f$ theorem

I'm studying Wiener's 1/f theorem, and I got curious about which was its original statement.I've been looking online but found nothing. I want to know if Wiener also proved the $n-$dimensional ...
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1answer
92 views

Fourier Series for $f(x)=\sin(x)+\cos(2x)$

Find the Fourier Series for $$f(x)=\sin(x)+\cos(2x)$$ I got $a_0=0$ which seems correct but I'm struggling with $a_n$ and $b_n$. Here are my attempts: $$\begin{align} a_n&=\frac{1}{2\pi} \...
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Can a well-behaved, positive-definite function $\phi(x)$ always be represented by $\phi(x)=\sum_{n=0}^{\infty}\mid a_{n}\psi(x)_n\mid^{2} $?

It's well known that some $\phi(x^{\mu})$ fulfilling a basic set of criteria (on a manifold M of arbitrary dimension lets say dim=1 here) may be represented as a fourier series of orthogonal ...
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14 views

Fourier Transform Spectrometer

Firstly, I was planning on constructing a Fourier Transform Spectrometer for a Physics project at school. Is this feasible? If so, what components could I use to construct it? Secondly, how exactly, ...
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2answers
140 views

$\lim_{\lambda \to \infty} \int^b_0 f(t) \frac{\sin(\lambda t)}{t} $

For a continuous function, $f:[0,b] \to \Bbb{R}$ show that: $$ \lim_{\lambda\to\infty} \int^b_0 f(t) \frac{\sin(\lambda t)}{t}\,dt = \frac{\pi}{2}\,f(0) $$ I know it has something to do with the ...
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$A_{mn} = \frac{1}{\pi}\int_0^{\pi}d\theta\, \sin(2m\theta)\, \frac{1-\cos^{2n}(\theta)}{\tan(\theta)} = $ ? $m$, $n$ integers > 0

The integral $$ A_{mn} = \frac{1}{\pi}\int_0^{\pi}d\theta\, \sin(2m\theta)\, \frac{1-\cos^{2n}(\theta)}{\tan(\theta)} $$ popped up when I was playing around with the integral representation of the ...
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Finding converges or not using Fourier series

$F=(9x+7\pi)$, given period $[-\pi;\pi)$. Find Fourier series for the given function if $x=-\pi$? Firstly I found $a_0$ which was equal to $7\pi$ and $a_k$ and $b_k$ were equal to $0$. So I found if $...
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49 views

How to find the system transfer function corresponding to a two dimensional matrix of optical transfer functions?

I would like to find the system transfer function corresponding to a two dimensional matrix of optical transfer function where: Each of the 3 times 5 = 15 interferometers produce 15 sets of ...
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2answers
63 views

The rate of convergence of the Fourier series near the discontinuity of a function

I'm trying to understand how bad the Gibbs phenomenon is. Say I have for instance a square wave on $[0, 1]$, and I want to approximate it with a Fourier series. How many terms would I need to ...
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Representing a positive definite function as the absolute value squared of a fourier series?

For the closed surface of some (pseudo)Riemannian manifold M , we're interested in a function $\rho$ which satisfies: $$m=\int_{\partial M}\rho(x,y,z) \, d\mathrm{vol}$$ Where $dvol$ is the ...
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1answer
28 views

Complex Fourier series and half-range expansions

I need to find the complex Fourier series for $f(x) = x$, where $0 < x < 2\pi$. I tried to solve this in two different ways, first with even extension, and then with odd, but I did not get the ...
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Fourier Transform: Musical Instruments cotd.

Upon analysing the Fourier Transform of a musical sound, are there any other applications of the Fourier Transform so obtained? Any ideas would be appreciated. Edit 1: To clarify the situation, I ...
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Extracting the sum from the Fourier series and expressing phase change

From the Fourier series of 3 basic wave forms, I need to extract the sum and the coefficient of the Sin function in order to input them in a code for plotting. The constants from the formulas don't ...
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Showing Following Fourier series converges to sawtooth function

This question is originated from S/S Fourier Analysis Chapter 2 Exercise 8. Problem says show sawtooth function$$ f(x)= \begin{cases} -\frac{\pi}{2}-\frac{x}{2}, -\pi<x<0\\ \frac{\pi}{2}-\frac{...
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Fourier series Coefficients and wolframalpha

1) Please can my answers be checked, including my final Fourier series. 2) Is it possible to use Wolframalpha to check my answers? If so, how will I go about doing this? Deduce the Fourier series ...
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1answer
43 views

Fourier Transform: Musical instruments

How do I Fourier Analyse the music produced by a musical instrument? What I mean is that what tools/applications are best suited to Fourier Analyse waves from musical instruments?
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1answer
37 views

How does shifting make this function odd?

[Supply current of 3 phase semi converter] I've been told that shifting this waveform left by 30 + a/2 will make it odd. Odd means f(a) = -f(a), right? So, how it that happening here? Or am i ...
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1answer
210 views

The Fejer kernel has this $\sin$ closed form.

Let $D_N$ be the $N$th Dirichlet kernel, $D_N = \sum_{k = -N}^N w^k$, where $w = e^{ix}$. Define the Fejer kernel to be $F_N = \frac{1}{N}\sum_{k = 0}^{N-1}D_k$. Then $$F_N = \frac{1}{N}\frac{\sin^...
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1answer
475 views

Convolution: $ f (-)*g = g(-)* f$ does this mean both $f$ and $g$ have to be even functions?

Assuming $f$ and $g$ are different functions, does $ f (-)*g = g(-)* f$ mean both $f$ and $g$ have to be even functions? In fact, this is equivalent to $f\star g = g \star f$ (i.e., cross-correlation ...
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1answer
88 views

Proving that the limit of an integral of a series exists

The goal is to show that the following limit exists $$\lim_{T\to\infty} \frac{1}{T}\int_{-T}^T f(x)dx$$ where $$f(x)=\sum_{n=1}^\infty \frac{e^{ia_n x}}{n^2}$$ I already showed that $f$ is bounded ...
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1answer
88 views

Can I have an analytical solution for $\frac{\partial \theta}{\partial t}=\frac{\partial^2\theta}{\partial {x}^2}+1$

Subjected to the following boundary conditions: $\left.\frac{\partial \theta}{\partial x}\right|_{x=0,t}=c_1\theta\bigg\vert_{x=0,t}$ $-\left.\frac{\partial \theta}{\partial x}\right|_{x=1,t}=c_2\...
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2answers
110 views

If $\int_0^{2\pi} q = 0$, then $\lim_{n \to \infty} \int_0^{2\pi}p(x)q(nx) \, dx= 0$

I'm learning about Fourier series and need help with the following exercise: Let the functions $p, q \in L^1([0, 2\pi])$ be bounded and $2\pi$-periodic. If $\int_0^{2\pi} q = 0$, show that $...
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Discrete Fourier transorm vs Discrete Fourier series

I am confused how the two related. They seem to be very related but I can't make a clear connection. Moreover, I can't make a connection for the case of I have a non-periodic function. What if one ...
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1answer
32 views

Pointwise convergence of Fourier series in two dimensions

By Carleson's Theorem, we know that for every $f\in L^2(\mathbb{T})$ $$ f(x)=\lim_{N\rightarrow\infty}\sum_{k=-N}^N\hat{f}(k)e^{2\pi ikx}\;\text{ a.e.} $$ Suppose now that $f\in L^2(\mathbb{T}^2)$. ...
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1answer
42 views

Using Parseval's Identity to estimate the integral of a function.

We can expand the cosine function in the interval $[0,\pi]$ in a sine series, obtaining an odd, $2\pi$-periodic extension of $cos (x)$. Let $f$ be that function, and define $g(x) = \int_{0}^{x}f(t) {\...
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34 views

Fourier series from wikipedia

How did this page come up with the Fourier coefficients? It basically jump after coming up with this $$A_n=\sqrt{a^2_n+b^2_n} , \quad\phi_n=\arctan\left(\frac{a_n}{b_n}\right).$$
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Show that $L^2(0,1)=\operatorname{span}\{e^{2\pi inx}\}_{n\in \mathbb Z}$

Q1) In a course, it's written that $L^2(0,1)$ is spanned by $\{e^{2\pi inx}\}_{n\in\mathbb Z}$. How can I show it ? Q2) Let $f\in L^2(0,1)$. Then we have Parseval equality, i.e. $$\left\|f\right\|^2=...
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41 views

Solving telegrapher's partial differential equation

Using the method of separation of variables and writing $u(x,t)=M(x)N(t)$, we can solve the equation $$u_{tt}-\gamma^2 u_{xx} + 2\alpha u_t=0$$ $$0<x<l$$ $$t\ge0, \alpha>0 \text{ (}{\alpha} \...
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1answer
44 views

Ways to justify this interchange of summation and integration

In evaluating this integral: $$\int_0^\infty \frac{\Im{\left(e^{e^{ix}} \right)}}{x}\text{d}x$$ My means of evaluation was to expand the numerator of the integrand as a fourier series (a.k.a. Taylor ...
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1answer
30 views

Calculating integral value of Fourier series

Given fourier series: $$\mathrm{S}\left(x\right) = {3 \over \pi}\sum_{n = 0}^{\infty} {\sin\left(\left[2n + 1\right]x\right) \over 2n + 1}\,,\qquad \left\langle -\pi,\pi\right\rangle $$ Evaluate: ...
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1answer
57 views

How is the study of wavelets not just a special case of Fourier analysis?

As far as I can tell, "wavelets" is just a neologism for certain "non-smooth" families of functions which constitute orthonormal bases/families for $L^2[0,1]$. How is wavelet analysis anything new ...
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1answer
69 views

If $|\sum_{k=1}^{n}\frac{\sin(k\theta)}{k}|<M$, is it true that $|\sum_{k=1}^{n}\frac{\sin(k\theta)}{k^{3}}|<M$?

It's proven that $\sum_{k=1}^{n}\sin(k\theta)/k$ is uniformly bounded for all $\theta\in\mathbb{R}$ and all $n\geq 1$. So there exists a $M>0$ such that $$\left|\sum_{k=1}^{n}\frac{\sin(k\theta)}{k}...
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1answer
45 views

Convergence of a cosine series

Given that $$\sum_{n=1}^\infty a_n<\infty,$$ and that $$\lim_{n\to \infty}b_n=0$$ Is the series $$\sum_{n=0}^\infty a_nb_n^{-2}(1-cos(b_n))$$ necessarily convergent?
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1answer
23 views

Fourier Series of this function

Find the Fourier series of this function, only by using sine functions. This is not a homework, I'm just practicing different problems for an exam. I know that all coefficients, except b0 should be 0. ...