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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105
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5answers
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: $$f(x)=\...
35
votes
3answers
58k 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 ...
32
votes
4answers
2k 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 ...
28
votes
2answers
3k views

Did Joseph Fourier ever make a pure mathematical mistake?

Cited by "Imre Lakatos and the Guises of Reason" John David Kadvany, 2001: It is remarkable that the nineteenth century was a time of error for mathematics: not trivial oversights or amateur ...
26
votes
9answers
46k views

Real world application of Fourier series

What are some real world applications of Fourier series? Particularly the complex Fourier integrals?
26
votes
1answer
449 views

Show that $ \lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{k=0}^{n-1}e^{ik^2}=0$

TL;DR : The question is how do I show that $\displaystyle \lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}e^{ik^2}=0$ ? More generaly the question would be : given an increasing sequence of integers $(...
25
votes
2answers
587 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 ...
22
votes
1answer
426 views

Using Fourier Series to compute sums

I have just started learning the basics of Fourier series and have some doubts about it. I am aware that Fourier series can be used to compute infinite sums. For example, $\zeta(2)$ and $\eta(2)$ can ...
22
votes
1answer
764 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 $u=u(x,y)$: ...
21
votes
5answers
3k views

Why do Fourier Series work?

I would like to have an intuitive understanding of Fourier Series. I mean, I know the formulas: $$ f(t) =\frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos(n\pi tL)+\sum_{n=1}^\infty b_n \sin(n\pi tL) $$ And ...
20
votes
2answers
717 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 $a(n)=\frac{1}{\log n}$....
19
votes
1answer
1k 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
945 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 ...
15
votes
2answers
415 views

Is there a (deep) relationship between these various applications of the exponential function?

Here is a list of some applications of the exponential function. 1) The exponential mapping in Lie theory. I put this first because my intuition tells me that this must be the most fundamental, or ...
15
votes
1answer
322 views

Integral that arises from the derivation of Kummer's Fourier expansion of $\ln{\Gamma(x)}$

I am trying to prove that for $0<x<1$, $$\color{blue}{\ln{\Gamma(x)}=\frac{1}{2}\ln(2\pi)+\sum^\infty_{n=1}\left\{\frac{1}{2n}\cos(2\pi nx)+\frac{\gamma+\ln(2\pi n)}{n\pi}\sin(2\pi nx)\right\}}$$...
14
votes
1answer
2k 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
3answers
423 views

Why is $\sum_{n=-\infty}^{\infty}\exp(-(x+n)^2)$ “almost” constant?

I did some numerical approximation of $$\sum_{n=-\infty}^\infty \exp(-(x+n)^2)$$ and found that this function is "almost" constant ($\approx 1.772$). Why does the sum fluctuate little? Is there a ...
13
votes
2answers
4k 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) = \sum_{n=...
13
votes
5answers
1k 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
349 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 $$\left|\sum_{k=2001}^{m}a_{k}\sin{(kx)}\...
12
votes
7answers
562 views

Why does $\sum_{k=1}^{\infty}\dfrac{{\sin(k)}}{k}={\dfrac{\pi-1}{2}}$?

Inspired by this question (and far more straightforward, I am guessing), Mathematica tells us that $$\sum_{k=1}^{\infty}\dfrac{{\sin(k)}}{k}$$ converges to $\dfrac{\pi-1}{2}$. Presumably, this can ...
11
votes
4answers
4k 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 $\sum_{n=1}^{...
11
votes
1answer
4k 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. ...
11
votes
1answer
2k 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 ...
11
votes
3answers
1k 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 ...
11
votes
3answers
2k views

Fourier series of function $f(x)=0$ if $-\pi<x<0$ and $f(x)=\sin(x)$ if $0<x<\pi$

$$f(x) = \begin{cases}0 & \text{if }-\pi<x<0, \\ \sin(x) & \text{if }0<x<\pi. \end{cases}$$ My attempt: I went the route of expanding this function with a complex Fourier series. ...
11
votes
1answer
131 views

$L^p$ norm of certain trigonometric polynomials bounded by $L^p$ norm of second derivative

Let $f$ be a trigonometric polynomial on the circle $\mathbb{T}$ with $\hat{f}(j) = 0$ for all $j \in \mathbb{Z}$ with $\lvert j \rvert < n$. I want to show $$ \|f''\|_p \geq Cn^2 \| f \|_p. $$ for ...
10
votes
3answers
397 views

Showing that $\sum_{k=0}^{\infty} a^{k} \cos(kx) = \frac{1- a \cos x}{1-2a \cos x + a^{2}}$ without using complex variables

The identity $$\sum_{k=0}^{\infty} a^{k} \cos(kx) = \frac{1- a \cos x}{1-2a \cos x + a^{2}} \ , \ |a| <1$$ can be derived by using the fact that $ \displaystyle \sum_{k=0}^{\infty} a^{k} \cos(kx) ...
10
votes
7answers
4k 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 : $$\sum_{n=1}^\infty \frac{\cos nx}{n}$$ ...
10
votes
4answers
458 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}} \int_0^...
10
votes
3answers
3k 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
1answer
235 views

Translation of a certain proof of $(\sum k)^2 = \sum k^3 $

In the question: Geometrical interpretation of $(\sum_{k=1}^n k)^2=\sum_{k=1}^n k^3$ the mysterious last answer involves concepts from electrical engineering and signal processing: Consider the ...
10
votes
0answers
175 views

About the Gibbs phenomenon for the Legendre polynomials as an orthogonal base of $L^2(-1,1)$

In a recent question, I proved that the Fourier-Legendre expansion of the function $f(x)=\text{sign}(x)$ over $(-1,1)$ is given by: $$2\sum_{m\geq 0}\frac{4m+3}{4m+4}\cdot\frac{(-1)^m}{4^m}\binom{2m}{...
9
votes
8answers
1k 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
1answer
200 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 $\left(\frac{k}{d}\...
9
votes
2answers
1k 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 ...
9
votes
2answers
157 views

$L^{2}$ Approximation Error of Fourier Series of Union of Disjoint Arcs

Given $N$ disjoint arcs $\{I_{\alpha}\}_{\alpha=1}^{N}\subset\mathbb{T} $,set $f=\displaystyle\sum_{\alpha=1}^{N}\chi_{I_{\alpha}}$ show that $$\sum_{|v|>k}|\hat{f}(v)|^2\le\dfrac{CN}{k}$$ This ...
9
votes
2answers
172 views

Why is periodic harmonic analysis only possible with sines?

This paper shows that if we consider odd functions on $(-\pi,\pi)$ in $L_2$, then the only $2\pi$-periodic function $f$ for which $f(nx)$ is a complete orthogonal system is the sine function. I'll ...
9
votes
0answers
114 views

Which Fourier series are “legal”?

Let's consider a continuous function $f(x)$ and real numbers $\lambda_n=(\alpha+\beta n)\pi$ where both $\alpha$ and $\beta$ are integers. In any interval $I$, is it true that $$ f(x)=\sum_{n\geq 0}...
9
votes
0answers
252 views

Interesting Integral with Parameters

I would like to compute the following integral: $$\int\frac{d^{2}\overrightarrow{q}}{2\pi}\int\frac{d^{2}\overrightarrow{k}}{2\pi}e^{i\overrightarrow{q}\cdot\overrightarrow{r}}\left(e^{i\...
8
votes
3answers
360 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 ...
8
votes
5answers
201 views

How to evaluate $\int_{0}^1 {\cos(tx)\over \sqrt{1+x^2}}dx$?

mean to calculate the integral $$\int_{0}^1 {\cos(tx)\over \sqrt{1+x^2}}dx$$ I have tried to consider the integral as fourier coefficient of $f(x)=1/\sqrt{1+x^2}$, however no ideas. Besides, let $F(...
8
votes
3answers
868 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
2answers
551 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 = \dfrac{\...
8
votes
8answers
2k 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
6answers
8k 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
9k views

Evaluate $\sum^\infty_{n=1} \frac{1}{n^4} $using Parseval's theorem (Fourier series)

Evaluate $\sum^\infty_{n=1} \frac{1}{n^4} $using Parseval's theorem (Fourier series). I have , somehow, to find the sum of $\sum_{n=1}^\infty \frac{1}{n^4}$ using Parseval's theorem. I tried some ...
8
votes
1answer
264 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 ...
8
votes
3answers
341 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 ...
8
votes
3answers
99 views

Is there a direct method for evaluating this integral: $\int_{0}^{2\pi}\ln^2(2\sin(\frac{x}{2}))dx$?

I stumbled upon this integral while attempting to evaluate $\sum_{n=1}^{\infty}\frac{\cos(n\theta)}{n}$. I started with the series $-\ln(1-z)=\sum_{n=1}^{\infty}\frac{z^n}{n}$, replaced z with $e^{i\...