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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0answers
331 views

Is there a particular meaning to the sum of Fourier coefficients $a_{n^2}$?

The formula $$\sum_{n=-\infty}^{+\infty} e^{-in^2x}$$ does not converge in any function space but it is perfectly valid in $\mathcal{D}'(\mathbb{R})$. When applied on a test function $\psi(x) = ...
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5answers
2k views

The Fourier series of $\sin^3 t$ in trigonometric form

I'm trying to calculate the Fourier series of $\sin^3t$ in trigonometric form. In previous excercises I have been able to use trigonometric identities to be able to calculate the coefficents, but here ...
3
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0answers
69 views

Is harmonicity preserved when taking limits (normal convergence) on the unit disk.

I'm reading Koosis's book on $H^p$ spaces and have a question. He is proving a $L^p$ version of the Dirichlet problem which states that if $F(t)$ is in $L^p$ on the unit circle then $$ ...
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2answers
1k views

Differentiating a triangular wave

I was really stuck and tried many times to differentiate the following series, and tried to convince myself that the differential form of a triangular wave is the square wave. But I couldn't work it ...
2
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1answer
65 views

Convergence of eigenmodes of a Sturm Liouville operator.

Is there any "eassy to see" proof for: "The eigenmodes of a Sturm Liouville ODE in a closed interval [a,b], with given boundary conditions, form a complete, orthogonal basis for continuous functions ...
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1answer
97 views

Exponential Fourier Series.

Determine the exponential Fourier series(which invovle exp(jkwt) terms) of the following. x(t)=cos(t)+cos(2t)+0.5 I calculated C0 and got the following. C0=0.5 however, I calculated Cm to be 0 ...
12
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1answer
332 views

Prove $\left|\sum_{k=2001}^{m}a_{k}\sin{(kx)}\right|\le 1+\pi $ ,$m\ge 2001,x\in R$

let $\{a_{n}\}$ is non-increasing postive sequence;show that if for $n\ge 2001,na_{n}\le 1$, then for any positive integer numbers $m\ge 2001,x\in R$, we have ...
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1answer
175 views

Inverse Fourier transform to find out $\hat c_1$

If we have an integration which is need to solve inversely $$a_0 e^{-r^2/R^2} = \int_0^\infty \hat{c}_1(k) \frac{\sin(k r)}{r} dk,$$ If I transform the $\sin(kr)$, then we get imaginary part. Please ...
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1answer
130 views

Fourier integral representations using only cosine functions.

Hi I have a question about Fourier integrals. Can Fourier cosine integrals represent any function, or just even functions?
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3answers
2k views

Find Fourier Series of the function $f(x)= \sin x \cos(2x) $ [duplicate]

Find Fourier Series of the function $f(x)= \sin x \cos(2x) $ in the range $ -\pi \leq x \leq \pi $ any help much appreciated I need find out $a_0$ and $a_1$ and $b_1$ I can find $a_0$ which is ...
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1answer
158 views

Fourier Transform and amplitude of waves

Given this definition of the fourier transform: $$f(t) \rightarrow \hat{f}(\omega)=\int\limits_{-\infty}^{+\infty}f(t)\,e^{-i\omega t}\,dt$$ and now ...
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1answer
46 views

Positive functions with negative Fourier tail

As the title indicates, my question is: Question: Does there exist a nonnegative function $f\in L^1(\mathbb R)$ such that the Fourier transform of $f$ satisfies $$\hat f(\xi)<0$$ for all ...
5
votes
1answer
105 views

Uniform boundedness of Fourier series of indicator functions

Suppose $f\in L^1[0,2\pi]$, denote by $S_n f(x)$ the partial sum of the Fourier series of $f$. I am interested in whether $S_nf(x)$ is uniformly bounded independent of $x$ and $n$, i.e. $$(*)\ \ \ \ ...
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1answer
399 views

Fourier Series of $f$ on the given interval

my goal is to find the Fourier series of f on the given interval: $$f(x) = \begin{cases} 0, & \text{if } -\pi < x < 0 \\ \sin(x), & \text{if } 0 \le x < \pi \end{cases}$$ I know ...
0
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2answers
224 views

Conceptual question about Discrete Fourier Transform

On the wikipedia page for the discrete Fourier transform, the first sentence says: In mathematics, the discrete Fourier transform (DFT) converts a finite list of equally spaced samples of a ...
5
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3answers
360 views

Further studies on Fourier Series and Integrals.

If you had to choose two books from the following list, which pair would you chose, and why? If you haven't read any, would you pick any pair among the list based on the author of the book? I am ...
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1answer
229 views

generating a random periodic function with bounded amplitude and bounded fourier coefficients

I would like to generate (i.e. repeatedly compute via a computer) a random periodic function $f(x)$ with period $T$ such that $|f(x)| \leq M$ and the kth Fourier coefficient $|A_k| \leq g(k)$ for a ...
3
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1answer
97 views

A PDE problem about $(\partial_x^2 + \partial_t^2)u = 0$ using Fourier series.

I'm trying to solve the following initial value problem (from Folland, pg. 277, exercise $48$a) using Fourier series: Let $x \mapsto f(x)$ and $x \mapsto u(x, t)$ be periodic functions on ...
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0answers
39 views

calculation wave function

I have a bunch of points from a segment (~1.5 periods) of a wave. The wave looks like a cosinus wave, but it isn't. The length between the left maximum and the minimum is shorter than the length ...
1
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0answers
358 views

Fourier-Bessel series coefficients

When finding the coefficients of a Fourier-Bessel series, the Bessel functions satisfies, for $k_1$and $k_2$ both zeroes of $J_n(t)$, the orthogonality relation given by: $$\int_0^1 ...
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0answers
187 views

Prove: $\int_0^{\infty}\left(\frac{\sin x}{x}\right)^2dx=\pi/2$

I am dealing exercise 12 in Chapter 8 of Rudin's Principles of Mathematical Analysis. Given the function $f$: $$f(x) = \begin{cases} 1, & \text{if $|x|\le\delta$} \\ 0, & \text{if ...
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1answer
342 views

Even and odd functions using integrals

If $f: [-r, r] \to\mathbb{R}$ is an even function, show that $g(x) = \cos(nx)$ is an even function and $h(x) = \sin(nx)$ is an odd function. Consider: $\int_{-r}^{r} f(x)\cos(nx)dx = 2\int_{0}^{r} ...
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2answers
733 views

Programming discrete fourier coefficients in matlab

Alright so I am having the following issue: I want to figure out how to find the fourier coefficients of the following function: $$D(X)=\frac {a'(x)} {1+a'(x)^2}$$ Where $a(x)$ is an arbitrary ...
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1answer
122 views

Is this infinite series a Fourier series?

I have what looks like a Fourier series but I don't quite understand how (or if) it is possible to recover a function from this. ...
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1answer
101 views

Finding the fourier series representation for a piecewise function

Expand the given function in the appropriate Fourier series: $$\begin{align} f(x) = \begin{cases} x+1 &\mbox{if } -1 \leq x \leq 0 \\ x-1 &\mbox{if } 0 \leq x \lt 1 \end{cases} \end{align}$$ ...
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votes
2answers
92 views

What is the notation of 'a single term in the DFT'

I have a notation/terminology question: I am writing a paper in not-quite-my-area and can't figure out the right way to phrase/notate the following: I have a discrete function $p[x]$, of which I can ...
2
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1answer
157 views

Why is it so difficult to prove that the discrete Fourier transform (DFT) cannot be calculated in faster time than $N \log N$?

As the title says, why is it so difficult to prove that the discrete Fourier transform (DFT) cannot be calculated in faster time than $O(N \log N)$? This is a famous open problem in ...
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1answer
69 views

Fourier series convergence in $L^2$

Consider a function $g \in L^2(-\pi,\pi)$ such that it is continuous at $x \in (-\pi,\pi)$. Prove that if the Fourier series of g converges at x then that implies g(x) is its limit. I was thinking ...
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1answer
67 views

What is the formula for complex fourier series?

I am watching this video on complex Fourier Series where the instructor states the formula as: $$ f(x) = C_0 + \sum_{-\infty}^{\infty}C_ne^{inx} $$ where as the notes on the same topic by ...
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0answers
48 views

Showing that $\mathrm{P}(t,x) = \sum_{n\in\mathbb{Z}} \mathrm{G}_t(x-2\pi n)\in\mathbb{C}^\infty((0,\infty)\times\mathbb{R})$

Welcome everybody :) I need your help in answering the following question: Let $t > 0$ and $\mathrm{G}_t(x) = (2\pi t)^{-1/2}e^{-x^2/2t}$ Show that $$\mathrm{P}(t,x) = \sum_{n\in\mathbb{Z}} ...
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0answers
74 views

how a discontinuous function converges to Hermite- Fourier series?

I have the proof using a text that if a function $f (x)$ is square integrable with weight function $e^{-x^2}$ and also is piecewise continuous, then $f (x)$ converges to ...
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2answers
61 views

Confused between multiple representations of Fourier Series' formula

I have never used the formula for Fourier Series with the representation that the instructor of the above video is using that involves $k$ and $\omega$. Instead, I use $n$ and $\pi$. Now, suppose ...
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1answer
465 views

Function not satisfying pointwise convergence and Fourier series

Can you show an example of a function that does not satisfy pointwise convergence theorem hypotheses for Fourier series but that is still expressible as Fourier series? [Added after comment] In ...
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1answer
649 views

Absolute convergence of Fourier series of a Hölder continuous function

Suppose that $f$ is $2 \pi$ periodic and Hölder continuous of order $\alpha > 1/2$. Show that the Fourier series of $f$ converges absolutely. So we know that $f(x+2 \pi t) = f(x)$ for all $t \in ...
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1answer
197 views

Complex form of Fourier Series

So, the last part of the university syllabus in the chapter of Fourier Series is: ...
0
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1answer
33 views

Am I understanding this integration right?

This is the snippet of a problem from this PDF here. What I dont understand is why they retain the $Sin$ part for evaluation after integration when all that it is going to evaluate to is 0. If I ...
0
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1answer
517 views

Some doubts regarding half range cosine series.

I am learning half range sine and cosine series of a given function. My understanding is this: If instead of a symmetric interval $[-L,L]$ you are provided by an interval $[0,L]$ then you will have ...
4
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1answer
1k views

Fourier Transform of spherical harmonics

I am trying seeking for definition (or some source) of the Fourier Transform of Spherical Harmonics (see https://en.wikipedia.org/wiki/Spherical_harmonics). Any help will be really appreciated. ...
0
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1answer
29 views

Trouble with fourier coefficients

I feel like this is a really simple question and that my brain is just not working. I am looking at this paper and I am looking at the T matrix defined under equation 20. The author says that ...
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1answer
58 views

Can we estimate the lower bound in this way?

This post is aimed to find a lower bound of $\sum_{k=1}^{n}\frac{\cos(kx)}{k}$ for arbitrary $n \geq 1$ ================================= My approach: Let $S_n(x)$ denote the partial sum of the ...
3
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1answer
118 views

Fourier series for $[x]-x+\frac{1}{2}$

$[x]-x+\frac{1}{2}$ has the Fourier series $$\sum_{n=1}^{\infty} \frac{\sin{2n\pi x}}{n\pi}.$$ By evaluating the series directly, which requires some work, it can be shown that the series is ...
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2answers
79 views

Calculating $a_0$ in Fourier Series

I am using this YouTube video to learn Fourier Series. The question can be clearly seen in the picture. The instructor calculates $a_0$ as the area under the triangle which is fine. Nothing wrong ...
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4answers
24k views

How does knowing a function as even or odd help in integration ??

So, I am learning Fourier Series and it involves integration. I am not too good at integration. Now, the resource I use is videos by Dr. Chris Tisdell. In the ...
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2answers
155 views

Laplace equation with weird boundary condition

So, guys, here's my problem. I have this differential equation $$ U''_{xx}+U''_{yy}=0 $$ with these boundary conditions: $$ U'_{y}(x,0)=0 $$ $$ U'_{y}(x,\pi)=0 $$ $$ U(0,y)=0 $$ $$ ...
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1answer
168 views

How can I proved, that $\left\{\sqrt{\tfrac{2}{\pi}}\sin(kx):k\in\mathbb{N}\right\}$ is an orthonormal basis of $L^2[0,\pi]$?

I want to prove that $S = \left\{\sqrt{\tfrac{2}{\pi}}\sin(kx):k\in\mathbb{N}\right\}$ forms an orthonormal basis of $L^2[0,\pi]$. I may use the fact, that $B = ...
2
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2answers
177 views

Question on Fourier series $\sum_{n=1}^\infty a_n \sin ( \pi nx) = f(x)$

Assume $\sum_{n=1}^\infty a_n \sin ( \pi nx) = f(x)$ where $f: [0,1] \rightarrow \mathbb R$ continuous is and $f(0) = f(1)$. Can I then recorver the $a_n$ by using somehow the Fourier series of $f$ ? ...
0
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1answer
256 views

Expansion in sine Fourier series

find the half range sine expansion Fourier series of the following function f(t)=t(π-t) 0≤t≤π find bn
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1answer
127 views

Determine coefficients of a Fourier series

Given the $2\pi$-periodic function $f(t)=t^2$ such that $-\pi \le t \le \pi$, I want to determine the coefficients $f_k$ of the fourier series of this signal. Therefore I use $$f_k = ...
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0answers
126 views

Can we compute Fourier series of any function this way?

There is a technique to compute Fourier series much quickly, but I doubt how general this technique can be. Let's look at a simple example to see how the technique goes. Compute Fourier series of the ...
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3answers
74 views

Help to compute the following coefficient in Fourier series $\int_{(2n-1)\pi}^{(2n+1)\pi}\left|x-2n\pi\right|\cos(k x)\mathrm dx$

$$\int_{(2n-1)\pi}^{(2n+1)\pi}\left|x-2n\pi\right|\cos(k x)\mathrm dx$$ where $k\geq 0$, $k\in\mathbb{N} $ and $n\in\mathbb{R} $. it is a $a_k$ coefficient in a Fourier series.