Questions on Fourier series, the expansion of a function in terms of basis functions that satisfy an orthogonality relation. Usually, complex exponentials/sines and cosines are used.

learn more… | top users | synonyms

3
votes
4answers
297 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
55 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
45 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
62 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
131 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
815 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
51 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
289 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
694 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
134 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
658 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
607 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
214 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
209 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
139 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
180 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 ...
85
votes
4answers
3k views

Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$

I would like to prove that $\displaystyle\sum_{\substack{n=1\\n\text{ odd}}}^{\infty}\frac{n}{e^{n\pi}+1}=\frac1{24}$. I found a solution by myself 10 hours after I posted it, here it is: ...
6
votes
2answers
437 views

How to expand the Fourier series for $f(x)=\max \{0, \frac{\pi}{2}-\lvert x\rvert \} $?

My Question: My Goal is to determine the Fourier series for $f(x)=\max \{0, \frac{\pi}{2}-\lvert x\rvert \} \quad$ for $x \in [-\pi, \pi ]$ This function is $2\pi$-periodic. My Approach: i found ...
0
votes
1answer
74 views

Determining Fourier series for $\lvert \sin{x}\rvert$ for building sums

My math problem is a bit more tricky than it sounds in the caption. I have the following Task (which i in fact do not understand): "Determine the Fourier series for $f(x)=\lvert \sin{x}\rvert$ in ...
2
votes
1answer
162 views

In my Fourier text book, there are the following exercises to prove. why do some of them have the same left side but have different right sides?

In my Fourier text book, there are the following exercises to prove.why do some of them have the same left side but have different right sides? The demand of these question is to prove these ...
2
votes
1answer
46 views

Need to find a Fourier Series…

I am to find a Fourier Series for the following function: $$ y(x)=\sqrt {R^{2}-x^{2}} $$ about $$ -R \leq x \leq R $$ with the recursion $$ y(x+2R)=y(x) $$ Do I let$\sqrt {R^{2}-x^{2}}$equal $y$ ...
5
votes
2answers
499 views

A Fourier series exercise

Can anyone give me a hand with this exercise about Fourier series? Let $f(x)=-\log|2\sin(\frac{x}{2})|\,\,\,$ $0\lt|x|\leq\pi$ 1) Prove that f is integrable in $[-\pi,\pi]$. 2) Calculate the ...
0
votes
0answers
214 views

Fourier series for $e^x$ over $[0,\pi)$

I am trying to solve the following, Find the Fourier series of $h(x) = \text{e}^x, x \in [0,\pi)$. I'm not sure how to approach it since the question does not specify whether to use an even or ...
0
votes
1answer
22 views

Question about a small part of this Fourier series problem

Calculate the Fourier series expansion for the following function of period 2: $f(t)=2+2t^2$ for $-1<t<1$ I just have a small question for this problem. I've already gotten $A_0$ to be ...
10
votes
4answers
418 views

Singular asymptotics of Gaussian integrals with periodic perturbations

At the bottom of page 5 of this paper by Giedrius Alkauskas it is claimed that, for a $1$-periodic continuous function $f$, $$ \int_{-\infty}^{\infty} f(x) e^{-Ax^2}\,dx = \sqrt{\frac{\pi}{A}} ...
1
vote
0answers
115 views

Intervals where the function is similar to the Fourier series

$$f(x)=\left\{\begin{array}{l l} 0,\quad x \in [-L,0[\\ 1,\quad x \in [0,L] \end{array}\right.$$ I need to know in which intervals the sum of the Fourier series is "equal to the function $f(x)$". ...
-1
votes
1answer
47 views

How to prove this Fourier question?

How to prove this Fourier question? I hope for a procedure in detail.
1
vote
0answers
187 views

Fourier Analysis of Prime Counting Function

I was thinking about the following: Denote $\pi(x)$ as the prime counting function such that: $$ \pi(x) = \#\text{ of prime numbers}\leq x $$ It is well known from the prime number theorem that $$ ...
1
vote
1answer
254 views

What is the odd Fourier extension of $\sin x \cos(2x)$?

odd half range extension of $$f(x) = \sin x \cos(2x)\text{ with limits $0$ to $\pi$}$$
2
votes
1answer
55 views

Convergence of Fourier series $\frac 1 {2i} \sum_{n \neq 0} \frac { \exp (inx)} n$

Let $$ f(x) := \begin{cases} -\frac \pi 2 - \frac x 2 && x \in (-\pi,0) \\ \frac \pi 2 - \frac x 2 && x \in (0, \pi) \\ 0 && x = 0 \end{cases} $$ I have to show that $\frac 1 ...
4
votes
1answer
8k views

What is the Fourier transform for $f(x)=e^{-x^2}$

I remember their being a special rule for this kind of function but I cant remember what it was. Anyone know how ? thanks
1
vote
0answers
41 views

Understanding the indices in a Fourier series

Sometimes the truncated Fourier series of a function with Fourier coefficients $\hat{u}_k$ is written $$\sum_{k=-N}^N\hat{u}_ke^{ikx}$$ which is a linear combination of $\cos(nx) +i\sin(nx)$ for ...
0
votes
1answer
52 views

Prove that the given sum is not Fourier series

Book browsing Banach spaces of Analytic function of the author Kenneth Hoffman on page 74 is one example. This example is compiled in this way: Prove that $$ \sum_{n=1}^{\infty}\frac{1}{\log ...
19
votes
1answer
668 views

Seeking Fourier series solution on Laplace equation…still looking, am I on track?

Okay, I've been working at this a couple of days now, I will try to give relevant details but will omit some intermediate steps. The problem as given says: Consider the BVP for $u=u(x,y)$: ...
2
votes
1answer
75 views

Computation of standard series

I am stuck on the computation of the following sum: $\sum_{k=1}^\infty e^{-n^2}\cos(n)$. Simple tricks fail and also i have no idea how to fit it for Fourier series. Are there any other ways?
2
votes
2answers
148 views

fourier series by lebesgue integral

hw: anyone knows how to find fourier series over the function $$ f(x)= \begin{cases} 1 & \text{if $x$ is irrational}\\ 0 & \text{if $x$ is rational} \end{cases} $$ by lebesgue integral? ...
0
votes
1answer
55 views

Show that Fourier coefficients approach zero uniformly

Let $f(t)$, $g(t)$ be piecewise continuous functons on $[-\pi,\pi]$, periodically continued on $\mathbb R$. I want to show that $$ a_n(x) = \frac{1}{\pi} \int\limits_{-\pi}^{\pi} f(x+t)g(t) ...
1
vote
2answers
4k views

Calculate the Fourier transform of ${\rm b}\left(x\right) = 1/\left(x^{2} +a^{2}\right)$

I need help to calculate the Fourier transform of this funcion $${\rm b}\left(x\right)=\frac{1}{x^{2} + a^{2}}\,,\qquad a > 0$$ Thanks.
0
votes
2answers
526 views

Trapezoid rule over trigonometric polynomials

The question is regarding trapezoid rule applied on trigonometric polynomials Here is the question Show that the composite trapezoid rule over an equidistant partitioning with interval size $h = ...
0
votes
1answer
89 views

Wave-Function Series?

So I was basically exploring the function: $\displaystyle {\text{frac}(x)}$ which is the fractional part function and I noticed that it has a nice fourier series definition which is: ...
1
vote
1answer
391 views

Fourier series of square wave with DC component (mean component) - amplitude question

Do I subtract the DC component (mean value) from the amplitude in my sine terms? $f(t)=\left\{ \begin{array}{l l} 0 & \quad -5\le t\leq 0\\ 1 & \quad 0< t\leq 5 \end{array} ...
0
votes
1answer
73 views

Fourier transforms - don't understand this concept!!! Please help me on this

I have two Fourier transforms to solve, but the problem is that a I have a characteristic bijection or some etching that I don't know what it is and I don't know how to solve this... Please help ...
2
votes
2answers
1k views

Sum over cosines = dirac delta - how to get the coefficients?

Given this formula: $$\sum\limits_{n=0}^\infty a_n \cos(n \pi x / d) = \delta(x-x_0)$$ Where $0 \leq x \leq d$. How can one calculate the coeffciients $a_n$? I googled and searched all kinds of ...