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.

learn more… | top users | synonyms

0
votes
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 ...
0
votes
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
votes
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 ...
1
vote
2answers
77 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 ...
1
vote
4answers
23k 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 ...
3
votes
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: $$ U'_{y}(x,0)=0 $$ $$ U'_{y}(x,\pi)=0 $$ $$ U(0,y)=0 $$ $$ ...
2
votes
1answer
166 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
votes
2answers
176 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
votes
1answer
250 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
1
vote
1answer
126 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 = ...
1
vote
0answers
125 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 ...
1
vote
3answers
72 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.
0
votes
1answer
155 views

Find Fourier Series coefficients of x=1 line function.

I want to know that can we find the Fourier series coefficients of the periodic signal x=1 where the limits are from -infinity to +infinity. The problem arises with the limits and it will converge to ...
1
vote
1answer
215 views

Find the fourier series of the following function

"Compute the Fourier series of the periodic function $f(x)$ that is defined in $\mathbb R$ as follows: $$f(x) = |x-2n \pi| $$ for all $x$ s.t. $(2n-1)\pi < x <(2n + 1)\pi.$ Give the ...
1
vote
1answer
61 views

approximate $[0, 1]$ continuous function with 2d basis.

everyone. I've been thinking of this problem when reading papers about Fourier series. I think I can state my question as follows: in the interval $[0, 1]$, I want to approximate an unknown ...
1
vote
1answer
135 views

Uniform convergence of Fourier Series, how do I check it?

Let $f(x)=x(\pi-x)$, $x\in (0,\pi)$. The function satisfies the Dirichlet conditions so its Fourier series, $S_f$ converges pointwise to $f$. The definition of a Fourier series of $f$ on $[a,a+L]$ ...
1
vote
0answers
97 views

Looking for feedback on Taylor Maclaurin and Fourier series

The Problem: You encounter the following wave when examining a digital switching circuit. You need to create a mathematical model so that you can examine changes in the circuit’s behavior. ...
6
votes
3answers
145 views

State-of-art of the Discrete Fourier Transform

I would like to know what is the state-of-art in the research of the discrete Fourier transform. I have listed some questions to help answering, please add your own to make the list more ...
1
vote
1answer
102 views

Fourier Series for $|x|$

How can I calculate the Fourier series for |x| (where $x\in[−\pi,\pi]$) in the complex form? Thanks.
10
votes
1answer
3k views

Taylor Series and Fourier Series

Taylor series expansion of function, $f$, is a vector in the vector space with basis: $\{(x-a)^0, (x-a)^1, (x-a)^3, \ldots, (x-a)^n, \ldots\}$. This vector space has a countably infinite dimension. ...
0
votes
1answer
2k views

Create Fourier-Series of f(x) = x if 0 < x < Pi and 0 if Pi < x < 2*Pi

I tried the following to create the Fourier-series of the function: $$ f(x) = \begin{cases} x & 0<x<\pi \\ 0 & \pi < x < 2 \pi \end{cases}$$ This is what I tried: $$a_0 = ...
1
vote
0answers
14 views

Fittig N-D periodic functions

Besides fourier series, are there any stable way to fit N-D nonlinear functions that demonstrate some degree of periodicity, sigmoid neural networks works poorly in this domain.
5
votes
2answers
171 views

A continuous function on the circle with divergent Fourier series.

I'm currently reading the Fourier analysis book and I have learned that every continuous function on the circle can be uniformly approximated by trigonometric polynomials, by using Fejer kernel. ...
0
votes
1answer
72 views

Fourier series question

How do we know that a Fourier series expansion does exist for a given function $f(x)$? I mean, if $f(x)=x$ and we suppose that $x=a_1\sin(x)+a_2\sin(2x)$ with $-\pi\leq x \leq \pi$ the Fourier ...
2
votes
0answers
41 views

Inverting a discrete linear transformation

Consider the transformation from the set $\{a_n\}_{n=0}^N$ to the set $\{p_j\}_{j=0}^N$: $$ p_j = \sum_{n = 0}^Na_n\phi_n(x_j)$$ where $\{\phi_n(x)\}_{n=0}^N$ is a set of basis functions (linearly ...
0
votes
1answer
35 views

Fourier series identity

I need to prove that $\dfrac{a \sin(bx)}{1 - 2a \cos(bx) + a^2} = \sum_{n=1}^\infty a^n \sin(nbx)$ where $|a| < 1$. It seems that this can be proved by using Euler's formula identities for ...
1
vote
1answer
546 views

Fourier series of $|A\sin(wt)|$

I am having some trouble calculating the fourier series of $x(t)=|A\sin(wt)|$. I have thought that the period is $T'=\frac{T}{2}=\frac{\pi}{w}$ so the result that i ended up was $c[n]=\dfrac{-A}{\pi} ...
3
votes
4answers
327 views

Is a Fourier Series a continuous function?

My question relates to the properties of the Fourier series of a function, $f: \mathbb{R} \to \mathbb{R}$. I know from an elementary course in differential equations (for engineers) that, for all ...
0
votes
1answer
56 views

Fourier transform of periodic signal

I have a question that is similar to this one but slightly different. If I have discrete signal $$s(t) = \sum_k n_k \delta(t-kT_0),\quad k=0,1,\dotsc,$$ where $n_k$ are just some scalar numbers. What ...
1
vote
0answers
47 views

Approximation the function $f(t)=I_0(-rt)e^{-rt}$ with sum of Exponentials.

Consider the function $f(t)=I_0(-rt)e^{-rt}$ where $I_0(t)$ is modified Bessel’s function and $r>0$. I am looking for an approximation for the function with a sum of exponential functions in $t ...
1
vote
1answer
64 views

Fourier series representing a continuous function?

I am fairly sure the answer to my question is "No", so this is more of an affirmation/reference request question. Given a Fourier series $\sum\limits_{k \in \mathbb{Z}} a_k e^{kxi}$, we can view it ...
1
vote
1answer
1k views

How do I convert a complex Fourier series into a purely real one

I have a question that gives me a periodic function $f(x)$ and asks me to find the complex Fourier series (which I think I have done correctly) and then asks me to obtain from that the regular Fourier ...
0
votes
1answer
50 views

Quick Confirmation of Fourier series using trigonometric identities

The Fourier series expansion for $f(x) = \sin 5x \sin x$ is $\dfrac{\cos 4x - \cos 6x}{2}$? This makes sense as $f(x) = \sin 5x \sin x$ is made up of the product of two odd functions which equals an ...
1
vote
1answer
137 views

Fourier series representation of even and odd functions

I'm not sure where to begin on showing that a Fourier series of a periodic function that is neither odd or even can be represented by the sum of the cosine fourier series and sine fourier series. I ...
1
vote
0answers
895 views

Represent non-periodic functions in a Fourier Series like function

I have this question of whether it is possible to represent non-periodic functions in a form just like you would represent a periodic function through a Fourier series. I understand this question ...
2
votes
0answers
54 views

Representing series $f(t)= \frac{\pi c^2}{l^2} \sum_{n=1}^\infty \frac{ n }{\omega_n}\cos(\omega_nt)$ as a Dirac comb function.

Consider the function $$f(t)= \frac{\pi c^2}{l^2} \sum_{n=1}^\infty \frac{ n }{\omega_n}\cos(\omega_nt)$$ where $\omega_n= \sqrt{(\frac{n \pi c}{l})^2-(\frac{r_0}{2})^2}.$ If we neglect the term ...
2
votes
0answers
308 views

Fourier series for piecewise function

let $-\pi \leq x\leq \pi$ and $$f(x)=\begin{cases}-x-\pi, & \text{ if} -\pi \leq x\leq -\pi/ 2\\ \;\;\;x, & \text{ if } -\pi/2 \leq x\leq \pi/2\\ -x+\pi, & \text{ if } \pi/2 \leq x\leq ...
2
votes
2answers
3k views

Fourier series for $f(x)=(\pi -x)/2$

I need to find the Fourier series for $$f(x)=\frac{\pi -x}{2}, 0<x<2\pi$$ Since the interval isn't symmetric over $0$, I guess I need to consider $f$'s periodic extension to $\mathbb R$. let's ...
1
vote
1answer
43 views

About mulit-variate Fourier series

If $f(x,y)$ is $2\pi$ periodic with respect to $x$ and $2\pi$ periodic with respect to $y$ respectively, then can I write $$ f(x,y) = \sum_{j,k \in \mathbb Z} c_{jk} e^{ijx} e^{iky}$$ where $$ c_{jk} ...
3
votes
2answers
795 views

Proof that $1/\sqrt{x}$ is itself its sine and cosine transform

As far as I understand, I have to calculate integrals $$\int_{0}^{\infty} \frac{1}{\sqrt{x}}\cos \omega x \operatorname{d}\!x$$ and $$\int_{0}^{\infty} \frac{1}{\sqrt{x}}\sin \omega x ...
2
votes
1answer
45 views

Integration question verifying piecewise

I have the following question: from direct integration show $\displaystyle \int \limits_{-L}^{L} \cos({m πx\over L})\cos({nπx\over L}) \ dx = \begin{cases}0 & m \neq n \\ L & m = n \\ ...
3
votes
1answer
139 views

What are the Fourier series of the function?

What are the Fourier series of $f(x)$ where $x \in [-\pi, \pi]$ defined by $$f(n) = \begin{cases} 1, & \text{if $x \in$ [0,$\pi$)} \\ 0, & \text{if $x \in$ [$-\pi$,0)} \\ \end{cases} $$ ...
2
votes
1answer
714 views

Fourier representation for $\tan(x)$

Q: Which Fourier representation is suitable for $f(x) = \tan(x)$: Fourier trigonometric series, Fourier half-range expansion, or Fourier integral and why? Well I searched and found that: $\tan(x)$ ...
1
vote
1answer
52 views

summation of this series as $ x \to \infty $ ??

given the series for the Mangoldt function $ \Lambda (n) $ $$ f(x)= \sum_{n=1}^{\infty}\frac{\Lambda (n)}{\sqrt{n}}\cos(\sqrt{x} \log n+\pi /4) $$ if we truncate the series, can we say that ...
4
votes
3answers
621 views

Finding the Fourier Series of $\sin(x)^2\cos(x)^3$

I'm currently struggling at calculation the Fourier series of the given function $$\sin(x)^2 \cos(x)^3$$ Given Euler's identity, I thought that using the exponential approach would be the easiest ...
3
votes
1answer
222 views

Using Fourier series techniques to solve $x'' + 3x = 7$ with $x'(0) = x'(5) = 0$

$$x'' + 3x= 7$$ Given conditions $x'(0)=x'(5)=0$. I checked the list and I went through three books. I am doing intro to differential equations. I just don't know how to get the extensions... I was ...
2
votes
1answer
210 views

Fourier analysis questions

Can anyone give me a hand with the proof of this properties? Prove that: a) The linear span of the set $\left\{T_bh/b\in\mathbb{R}\right\}$ is dense in $L_2(\mathbb{R})$, where $h(x)=e^{-\pi x^2}$. ...
1
vote
1answer
91 views

Fourier analysis question, orthonormal basis.

I need some help with this exercise: Given $A>0$, let $L_{A}^2(\mathbb{R})$ the subspace of $L^2(\mathbb{R})$ of the functions $f$ that satisfy $\hat{f}=\chi_{[\frac{-A}{2},\frac{A}{2}]}\hat{f}$. ...
3
votes
1answer
150 views

Fourier series $\sum_{m=0}^\infty \frac{\cos (2m+1)x}{2m+1}$

Does anyone know the sum of Fourier series $$\sum_{m=0}^\infty \frac{\cos (2m+1)x}{2m+1}?$$ I tried WA; it does not return a function.
1
vote
1answer
191 views

Parseval's identity

How to prove the Parseval's identity , I know the formal way but how to justify the interchange between the integral and the sum in a rigorously way , in addition what extra condition does the ...