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.

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66
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4answers
2k 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: ...
22
votes
2answers
514 views

Is $\sin\left(\frac{\pi}{2}\sin\left(\frac{\pi}{2}\cdots\sin x\cdots\right)\right)=\frac4{\pi}\sum\limits_{k=0}^\infty\frac{\sin(2k+1)x}{2k+1}$?

We can see intuitively that $$ f(x)=\sin\left(\frac{\pi}{2}\sin\left(\frac{\pi}{2}\sin\left(\frac{\pi}{2}\cdots\sin{x}\cdots\right)\right)\right) $$ is the square wave with period $2\pi$ and has the ...
18
votes
3answers
1k 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 ...
18
votes
1answer
496 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 $u=u(x,y)$: ...
17
votes
1answer
676 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 ...
15
votes
3answers
847 views

Which Fourier series formula is correct

I'm getting started on Fourier series but I'm confused over the formulae involved. My lecturers notes, including Wikipedia state that, for the interval $(-\pi \le x \le \pi)$ $$a_0 = \frac1\pi \int ...
14
votes
1answer
903 views

Making use of Fourier series to evaluate an infinite sum

Show that $$\sum_{k=1}^{\infty}\frac{(-1)^{k-1}k \sin(ax)}{a^{2}+k^{2}}=\frac{\pi}{2}\frac{\sinh(ax)}{\sinh(\pi a)}, \;\ x\in (-\pi,\pi)$$ It appears to me this series is crying out for the use of ...
13
votes
5answers
3k views

What are some real world application of fourier series?

what are some real world application of Fourier series ? particularly the complex Fourier integrals
13
votes
5answers
769 views

Geometric intuition behind convergence of Fourier series

I've been trying to work out the best way to understand why Fourier series converge, and it's a little embarrassing but I don't even know a rigorous proof. Can someone please help put me on the right ...
13
votes
1answer
487 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 ...
12
votes
1answer
308 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 ...
11
votes
2answers
685 views

Why is this allowed? (“Fourier's Trick”; finding the coefficients in a Fourier Series)

In my textbook (Introduction to Electrodynamics, D. Griffiths), we derive the equation for some strange potential function. Eventually, we get to this (for $n \in \mathbb{Z}^+$): $$ V_0(y) = ...
10
votes
3answers
2k views

The mathematics of music - why sine waves?

Of course, the Fourier transform is an extremely elegant mathematical method of overwhelming simplicity, and this straight away puts sine waves (or complex exponentials) on a high pedestal. But what ...
10
votes
2answers
397 views

Fourier series of function $f(x) = \begin{cases}0 & \text{if }-\pi<x<0 \\ \sin(x) & \text{if }0<x<\pi \end{cases}$

$$f(x) = \begin{cases}0 & \text{if }-\pi<x<0, \\ \sin(x) & \text{if }0<x<\pi. \end{cases}$$ My attempt: I went the route of expanding this function with a complex Fourier series. ...
9
votes
7answers
464 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 ...
9
votes
4answers
342 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}} ...
9
votes
1answer
175 views

Is $\sum_{n=1}^{\infty}{\frac{\sin(nx)}n}$ continuous?

Considering the infinite series $\sum_{n=1}^{\infty}{\frac{\sin(nx)}n}$ , I can show that it is not convergent uniformly by Cauchy's criterion and that it is convergent for every $x$ by Dirichlet's ...
9
votes
1answer
190 views

Fourier Series involving the Jacobi Symbol

We know that the Fourier Series $$s(x)=\sum_{k\neq0}\frac{1}{k}\exp\left(2\pi ik x\right)$$ corresponds to the sawtooth function, $s(x)=\left\{x\right\} -\frac{1}{2}$. Suppose that ...
8
votes
5answers
14k views

Difference between Fourier series and Fourier transformation

Whats the difference between Fourier transformations and Fourier Series? As I've been working with Fourier Series in my maths lectures yet a friend of mine also doing engineering has been working with ...
8
votes
8answers
1k 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 ...
8
votes
3answers
486 views

Condition for Fourier series

I read that Any "well-behaved" function of period $2\pi$ can be expressed as a Fourier series. What qualifies as "well-behaved"? Any examples of functions that cannot be expressed as a ...
8
votes
3answers
227 views

Meaning/Justification for Describing Functions as 'Orthogonal'

When introducing Fourier series, my lecturer stated that 2 periodic functions, $f$ and $g$, with period $2L$ are orthogonal iff $$\int^{L}_{-L}{f(x)g(x)}\mathrm dx=0$$ Wikipedia agrees, even defining ...
8
votes
4answers
4k views

Fourier series for $\sin^2(x)$

I was asked to compute the Fourier series for $\sin^2(x)$ on $[0,\pi]$. Now this is what I did and I'd like to know if I'm right. $\sin^2(x)=\frac12-\frac12\cos(2x)$ . I got the right hand side using ...
8
votes
2answers
727 views

Gibbs Phenomenon

Can someone explain in complete detail with the appropriate convergence arguments of the Gibbs Phenomenon for Fourier Series? I know that the overshoot near a jump does not die out as the frequency ...
8
votes
1answer
113 views

$\lim_{n\to\infty}\int_{-\pi}^{\pi}f(t)\cos^2(nt) \,dt$?

Let $f \in C[-\pi,\pi]$. Find the following limit: $$\lim_{n\to\infty}\int_{-\pi}^{\pi}f(t)\cos^2(nt)\,dt\,?$$
7
votes
4answers
162 views

Evaluate $\int_{-\pi}^\pi \big|\sum^\infty_{n=1} \frac{1}{2^n} e^{inx}\big|^2 \operatorname d\!x$

I am trying to solve exercises for the coming exam, and I am stuck on this exercise: Evaluate $$\int_{-\pi}^\pi \Big|\sum^\infty_{n=1} \frac{1}{2^n} \mathrm{e}^{inx}\,\Big|^2 \operatorname d\!x$$ ...
7
votes
4answers
1k views

Use Fourier series for computing $\sum_{n=1}^{\infty}\frac{1}{(2n-1)^2}$

I need to compute Fourier series for the following function: $f(x)=\frac{-\pi}{4} $ for $-\pi \leq x <0$, and $\frac{\pi}{4} $ for $ 0 \leq x \leq \pi$, and then to use it and compute ...
7
votes
4answers
316 views

The leap to infinite dimensions

Extending this question, page 447 of Gilbert Strang's Algebra book says What does it mean for a vector to have infinitely many components? There are two different answers, both good: 1) The ...
7
votes
2answers
186 views

Compute $\sum\limits_{n = 1}^{\infty} \frac{1}{n^4}$.

Compute the Fourier series for $x^3$ and use it to compute the value of $\sum\limits_{n = 1}^{\infty} \frac{1}{n^4}$. I determined the coefficients of the Fourier series, which are $$a_0 = ...
7
votes
2answers
173 views

Does $\sum_{n=0}^\infty\frac{\sin(2n+1)}{2n+1}=0$?

I've come to a bit of a sticking point in my answer to problem 14A given here http://www.maths.cam.ac.uk/undergrad/pastpapers/2011/ib/List_IB.pdf (note that this is a past paper that I am trying for ...
7
votes
3answers
309 views

For what sequences of real numbers $\left\{ k_{n}\right\}$ is the set of functions $\left\{ e^{ik_{n}x}\right\}$ a basis?

It is well known that the set of functions $\left\{ e^{^{inx}}\right\}$, for integer $n$, is an othonormal basis for the space of square integrable real functions in the interval $[-\pi,\pi]$. Now ...
7
votes
1answer
227 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 ...
7
votes
1answer
639 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. ...
7
votes
1answer
69 views

Finding a function from a fourier series

Taken from Apostol Analysis, it says, find a continuous function that generates the fourier series: $$ \sum_{n} \frac{-1^n}{n^3} \sin(nx) $$ I really have no idea how to solve this, instinctively I ...
7
votes
1answer
200 views

Computing Fourier transform for $L^2$ function

For a function $f\in L^1(\mathbb{R})$, its Fourier transform is defined as $$\hat{f}(y)=\int_{-\infty}^\infty f(x)e^{-ixy}dx$$ For a function $f\in L^2(\mathbb{R})$, its Fourier transform is ...
6
votes
2answers
735 views

Prove: Fourier series of $e^{\cos x} \sin (\sin x)$ is $\sum_{n=0}^{\infty}\frac{\sin (nx)}{n!}$

I'd love your help with proving that the following series $$\sum_{n=0}^{\infty}\frac{\sin (nx)}{n!}$$ is the Fourier series of $e^{\cos x} \sin (\sin x)$. I tried to find $\hat f(n)$ using ...
6
votes
3answers
173 views

Why does the Fourier series of $x$ not seem to give the right value?

I'm reading a lecture about Fourier series , and it says that you can represent any continuous function as Fourier series. There's a given example: Let $f(x) = x$. $f(x) \approx ...
6
votes
1answer
650 views

Term by term differentiation

If a function $f(x)$ is expressed as a Fourier series and we know $f'(x)$. Is it then true that if we differentiate the Fourier expression we must get $f'(x)$? E.g. if $f(x)=x^2$ for $x\in ...
6
votes
1answer
242 views

Convergence of Fourier Series

Is there an $f\in L^1(\mathbb{T})$ whose Fourier series converges a.e. on $\mathbb{T}$ but not a.e. to $f$?
6
votes
1answer
212 views

About $2$-periodic continuous solutions of $f(x)+f(x+1)=f(2x+1)$

Suppose I want to find all the continuous solutions to the functional equation $$f(x)+f(x+1)=f(2x+1),\tag{E1}$$where $f$ is a continuous and $2$-periodic function defined on the dyadic rationals. I ...
6
votes
1answer
226 views

An elegant non-technical account on the work of Joseph Fourier.

It would seem difficult for a naive person to understand the beauty of work done by Fourier. So as far as I know, one can use the Fourier transforms, analysis and series to apply them for heat ...
6
votes
2answers
338 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 ...
6
votes
1answer
919 views

Proving a family of orthogonal functions is complete over a certain interval

I'm reading Applied Partial Differential Equations by DuChateu and Zachmann, and the first couple of chapters contain quite a bit of review of Fourier series, as well as theory about L2 integrable ...
6
votes
1answer
616 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 ...
6
votes
1answer
304 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): ...
6
votes
1answer
329 views

Brownian motion and Fourier series

Let $(B_t)_{t \in [0, \infty)}$ be a Brownian motion. Can you prove me why it can be written as $$B_t= Z_0 \cdot t + \sum_{k=1}^{\infty} Z_k \frac{\sqrt{2} \cdot \sin(k \pi t)}{k \pi}$$ for some ...
6
votes
1answer
345 views

A Fourier series failing to converge on the Cantor Set

This is a strengthening of Chandru's question: http://math.stackexchange.com/questions/2227/ Is there a nice and concrete example of a Fourier series that fails to converge on some "big" set of ...
6
votes
0answers
245 views

expansion for $1-|t|$

Let $f$ be a continuous function on $\mathbb{R}$ with compact support with exactly one maximum. Form the functions $$ f_{m,k}(x)=f^m\left(x-\frac{k}{2^m}\right) $$ I am wondering if one can expand ...
5
votes
2answers
690 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
5answers
2k 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 : $A_{0}+\sum_{n=1}^{\infty}(A_{n} \cos{nx} + B_{n} \sin{nx})$ ...