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.
2
votes
2answers
233 views
Complex Fourier series
I need to find the complex Fourier series of this function, and I'm having problems calculating these integers:
$$|a|<1$$
$$x\in [-\pi,\pi]$$
$$f(x)=\frac{1-a\cos(x)}{1-2a\cos(x)+a^2}$$
...
13
votes
3answers
731 views
Computing $\zeta(6)=\sum\limits_{k=1}^\infty \frac1{k^6}$ with Fourier series.
Let $ f$ be a function such that $ f\in C_{2\pi}^{0}(\mathbb{R},\mathbb{R}) $ (f is $2\pi$-periodic) such that $ \forall x \in [0,\pi]$: $$f(x)=x(\pi-x)$$
Computing the Fourier series of $f$ and ...
2
votes
1answer
189 views
Asymptotic error of Fourier series partial sum of sawtooth function
In Iwaniec's book, Topics in Classical Automorphic Forms, pg. 4, he gives the statement:
$$\{x\}=\frac{1}{2}-\sum_{n=1}^N\frac{\sin 2\pi nx}{\pi n}+O((1+||x||N)^{-1})$$
where $\{x\}$ denotes the ...
3
votes
1answer
148 views
How many ways to calculate: $\sum_{n=-\infty}^{+\infty}\frac{1}{(u+n)^2}$ where $u \not \in \Bbb{Z}$
Today I have encounter a series:
$$\sum_{n=-\infty}^{+\infty}\frac{1}{(u+n)^2}=\frac{\pi^2}{(\sin \pi u)^2}$$
where $u \not \in \Bbb{Z}$
. I have known a method to computer it (by Residue formula):
...
2
votes
1answer
162 views
Convergence of: $\sum\limits_{n=1}^{\infty} \frac{(-1)^{n+1}\sin(nx)}{n}$
Need help with checking: $\sum\limits_{n=1}^{\infty} \frac{(-1)^{n+1}\sin(nx)}{n}$
for point-wise convergence and uniform convergence of: ${-\pi} \leq x \leq {\pi}$.
1
vote
2answers
374 views
$S(x)=\sum_{n=1}^{\infty}a_n \sin(nx) $, $a_n$ is monotonic decreasing $a_n\to 0$: Show uniformly converges within $[\epsilon, 2\pi - \epsilon]$
$S(x)=\sum_{n=1}^{\infty}a_n \sin(nx) $, $a_n$ is monotonic decreasing $a_n\to 0$, when ${n \to \infty}$. I need to prove that for every $\epsilon >0$, the series is uniformly converges within ...
44
votes
3answers
787 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:
...
8
votes
8answers
715 views
Conceptual/Graphical understanding of the Fourier Series.
I've been reading about how the Fourier Series works, so like how the orthogonality cancels out all but the one that we're looking for. I've read derivations of the Fourier Series. What I would like ...
6
votes
1answer
518 views
Pointwise but not uniform convergence of a Fourier series
What is an example of a continuous, or even better, differentiable, $2\pi$ (or 1) periodic function whose Fourier series converges pointwise but not uniformly? (Such function cannot be of Hölder ...
4
votes
1answer
642 views
Differentiability and decay of magnitude of fourier series coefficients
I want to know the answer/references for the question on decay of Fourier series coefficients and the differentiability of a function. Does the magitude of fourier series coefficients {$a_k$} of a ...
4
votes
5answers
1k views
Example of a trigonometric series that is not fourier series?
My textbook doesn't give any example of this kind of series. Could you provide some?
Trigonometric series is defined in wikipedia as :
$A_{0}+\sum_{n=1}^{\infty}(A_{n} \cos{nx} + B_{n} \sin{nx})$
...
3
votes
4answers
199 views
How to solve Real Analysis
Find the Fouries series of the function given below
$$
f(x) = \lvert\cos t\rvert \quad\hbox{for all $t$}.
$$
What does it means for all $x$? Can someone show me some work done so that I can ...
3
votes
1answer
227 views
One more question about decay of Fourier coefficients
Let $$f=\sum f_{s}\exp(2\pi isx)\in C^{(p-1)}[0,1]$$ and $$f^{(p)}\ in\ L_2[0,1]\ \ ( \sum\left|f_{s}\right|^{2}j^{2p}<\infty )$$
Does it imply that $f_s=O(s^{-(p+\psi)})$ for some ...
1
vote
2answers
206 views
Wave equation with initial and boundary conditions - is this function right?
If $y(x,t)$ satisfies the 1-dimensional wave equation
$$\frac{\partial^2y}{\partial t^2}=c^2\frac{\partial^2y}{\partial x^2}\quad\text{for }0\leq x \leq l$$
with boundary conditions
...
5
votes
2answers
246 views
log sin and log cos integral, maybe relate to fourier series
I try to use the method of differentiation under integral sign for the first one
And integrate it back, but I failed to find the constant $c$ ....
Anyone hav other method?
$$\begin{align}
& ...
5
votes
1answer
161 views
For a polygon on complex plane, when are the vertex 'Fourier coefficients' non-zero
Consider an $n$-sided convex polygon $P$ that contains the origin in the complex plane. Let the $j$-th vertex be denoted $z_j = r_j e^{i\theta_j}$ ($0 \leq \theta_j < 2 \pi$) for $j= 1 \dots n$. ...
1
vote
0answers
57 views
Intervals where the function is similar to the Fourier series
$$f(x)=\left\{\begin{array}{l l}
0,\quad x \in [-L,0[\\
1,\quad x \in [0,L]
\end{array}\right.$$
I need to know in which intervals the sum of the Fourier series is "equal to the function $f(x)$".
...
9
votes
4answers
291 views
Singular asymptotics of Gaussian integrals with periodic perturbations
At the bottom of page 5 of this paper by Giedrius Alkauskas it is claimed that, for a $1$-periodic continuous function $f$,
$$
\int_{-\infty}^{\infty} f(x) e^{-Ax^2}\,dx = \sqrt{\frac{\pi}{A}} ...
17
votes
1answer
477 views
Accessible proof of Carleson's $L^2$ theorem
Lennart Carleson proved Luzin's conjecture that the Fourier series of each $f\in L^2(0,2\pi)$ converges almost everywhere. Also, Richard Hunt extended the result to $L^p$ ($p>1$).
Some time ago I ...
7
votes
1answer
206 views
When is the weighted space $\ell^p(\mathbb{Z},\omega)$ a Banach algebra ($p>1$)?
Let $\omega:\mathbb{Z}\to (0,\infty)$ and let $1\leq p<\infty$. Consider the space $\ell^p(\mathbb{Z},\omega)$ of complex valued sequences $f=(a_n)_{n \in \mathbb{Z}}$ such that
...
4
votes
1answer
259 views
Rate of Fourier decay of indicator functions
The Fourier transform of the indicator function of an interval
$$\widehat{\chi}_{[a,b]}(\xi)=\int^b_{a} e^{i \xi x}dx=\frac{e^{i\xi b}-e^{i\xi a}}{i\xi}$$
has decay $O(|\xi|^{-1})$ as ...
3
votes
1answer
206 views
Conditions for a finite Fourier series
Under what circumstances is the Fourier series of a function guaranteed to have a finite number of coefficients?
1
vote
3answers
124 views
A integral with polygamma
I was doing a integral, the last part is
$$\int_0^{\frac{\pi}{2}}x^3\csc x\text{d}x$$
I ran this on Maple, it turns into polygammas...How we evaluate this?
I think there should be a way to evaluate ...
6
votes
2answers
170 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 ...
5
votes
1answer
253 views
Cusp forms' Fourier coefficients sign changes
I need some clarification on the following, if possible: I have seen in that for every $ f \in S_k$ which Fourier transform is $\sum_{n=1}^\infty a(n)q^n$ there is an upper bound $\sum_{n=1}^N ...
3
votes
1answer
130 views
Heuristic\iterated construction of the Weierstrass nowhere differentiable function.
I'm very interested in finding a way or hint for the construction of the Weierstrass function which is everywhere continuous but nowhere differentiable - let's call this (ECND). My most humble example ...
3
votes
3answers
1k views
an example of a continuous function whose Fourier series diverges at a dense set of points
Please give me a link to a reference for an example of a continuous function whose Fourier series diverges at a dense set of points. (given by Du Bois-Reymond). I couldn't find this in Wikipedia.
2
votes
3answers
123 views
Calculating the Fourier series of $x^{3}$
I was given as homework to calculate the Fourier series of $x^{3}$.
I know, in general, how to obtain the coefficients of the series using
integration with $$\sin(nx),\cos(nx)$$ multiplied by the ...
2
votes
2answers
128 views
Another integral with Catalan
Show that:
$$\int_0^1\frac{\arcsin^3 x}{x^2}\text{d}x=6\pi G-\frac{\pi^3}{8}-\frac{21}{2}\zeta(3)$$
I evaluated this by some Fourier series. Is there any other method?
Start with substitution of ...
1
vote
5answers
174 views
Fourier Analysis
I am interested in Fourier Analysis. But I don't get why the coefficients are choosen that way and why the Fourier series will converge to a given function?
Can someone provide me simple information ...
1
vote
3answers
202 views
Can any continous,bounded function have a fourier series?
In particular,can an oscillatory function with some decay term ( i.e e^(-t)*cos(kt) have a fourier series representation? All the articles I read said that the function has to be periodic,but this one ...
1
vote
2answers
501 views
Fourier transform of $\mathrm{sinc}(4t)$
I'm preparing for an exam in the signals and systems class I'm taking. One of the practice exams has a problem that requires you to take the Fourier transform of $\text{sinc}(4t)$.
From a table of ...
8
votes
7answers
256 views
Why does this Fourier series have a finite number of terms?
I am learning about Fourier series in class and the basic form of a Fourier Series is
$$a_{0}+\sum_{n=1}^{\infty} [a_{n}\cos(nx)+b_{n}\sin(nx)]$$
so a fourier series should have an infinity number ...
5
votes
2answers
356 views
Relationship of Fourier series and Hilbert spaces?
I just read in a textbook that a Hilbert space can be defined or represented by an appropriate Fourier series. How might that be? Is it because a Fourier series is an infinite series that adequately ...
5
votes
1answer
266 views
Series which are not Fourier Series
How to show that
$$
\sum_{n=2}^\infty \frac{\sin{(nx)}}{\log n}
$$
not the Fourier series of any function?
I have shown that the series is convergent by Dirichlet test.
Let $a(n)=\frac{1}{\log ...
4
votes
2answers
526 views
The Fourier series $\sum_{n=1}^\infty (1/n)\cos nx$
The series $$\sum_{n=1}^\infty \frac{\sin nx}{n}$$ is the Fourier series of the odd
$2\pi$-periodic extension of $(\pi-x)/2, 0<x<\pi$.
My question is : $$\sum_{n=1}^\infty \frac{\cos nx}{n}$$ ...
3
votes
1answer
249 views
heat equation solution
This is the question, I have solved it but I need someone to double check my solution. Question: Find the temperature $u(x, t)$ in a rod of length $L$ if the initial temperature is $f(x)$ throughout ...
3
votes
2answers
612 views
Lipschitz Continuity and Hölder Continuity helps Fourier series to converge
Let $f$ satisfies
$$|f(x+u) - f(x)|\leq L|u|^{\alpha}$$
for some constants $L$ and $\alpha$. If $\alpha = 1$ then $f$ is called Lipschitz continuous, and if
$0 < \alpha < 1$ ...
3
votes
2answers
1k views
Fourier cosine transform
Find Fourier cosine transform of $e^{-a^2 x^2}$ and hense evaluate Fourier sine transform of $x\cdot e^{-a^2x^2}$.
I can solve this question only if there is $x$ instead of $x^2$ in the exponential ...
2
votes
2answers
65 views
How to solve this equation by Fourier series?
$$ y''+3y=\sin ^4 x ,\quad y=\frac{1}{8} +\frac{\cos2x}{2}-\frac{\cos4x}{104}.$$
Now the text book states the solution, but I don't know the process of solving this equation. I need your help!
2
votes
1answer
169 views
Fourier and Legendre series
Find the Fourier sin series for the function $f(x) = x^3$ on the interval $0\leq x \leq L$. the Legendre series for the same function. One representation involves an infinite number of terms, ...
2
votes
3answers
101 views
Series help, fourier series
How do I know if a given function can be represented by a fourier series, that converges to the value of that function at non discontinuities. Also where did Fourier come up with the idea of ...
2
votes
3answers
192 views
About completeness of the Fourier series.
The Fourier series of a function is given by $$ \frac{a_0}{2} + \sum_{n=1}^\infty a_n \cos n \theta + \sum_{n=1}^\infty b_n \sin n \theta . $$ Here what does the statement " $\sum_{n=1}^\infty b_n ...
2
votes
2answers
251 views
The link between vectors spaces ($L^2(-\pi, \pi$) and fourier series
So in my PDE course we started with a review of complex numbers and vector spaces to introduce us to fourier series. I have a few questions about this.
I know 'big ell 2' and 'little el 2' are ...
1
vote
0answers
69 views
Fourier Series on a 2-Torus
Taking into account the answer given to this question, in special, the relation between the eigenfunctions of the Laplace-Beltrami operator and the Characters of a group does this imply that on a ...
1
vote
1answer
301 views
Dirichlet kernel.
I have a function $h\in L^1(\mathbb{T})$, and I want to show that:
$$\int_{\pi\geq |t|>\delta>0} h(x+t)D_N(t) dt/2\pi \leq \xi_N(h,\delta)$$
where $\xi_N(h,\delta) \rightarrow 0$ as ...
1
vote
2answers
2k views
How to find inverse Fourier transform
I have the function
$$ \delta(f-2) $$
How can we inverse Fourier transform it? It's easy if $f$ is replaced with $w$. But based on my knowledge, $w = 2\pi f$.
The correct answer is
$$ e^{4\pi i ...
1
vote
3answers
1k views
Scaling property of Fourier series and Fourier Transform
This question about the intuition behind the scaling property of the Fourier transform made me wonder about the corresponding notion for a Fourier series.
The Fourier transform of $f(ax)$ is ...
1
vote
2answers
169 views
$f,g$ are two continuous functions with period$=1$, are the Fourier coefficients $f*g=f(n)g(n)$?
Let $f,g$ be two continuous functions with period$=1$.
Are the Fourier coefficients of $f*g$ are given by the products $f(n)g(n)$ (of the $n$-th coefficient in each series)?
Thanks!
1
vote
1answer
592 views
Convergence of Fourier series for $|\sin{x}|$
I was solving this question I saw in a textbook. The question is :
Calculate the Fourier series for $ f(x) = |\sin x| $ for $-\pi \leq x \leq \pi$.
Which I had $ f(x) = \frac{a_{0}}{2} + \sum ...
