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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29
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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 ...
3
votes
3answers
4k 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.
86
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: ...
3
votes
1answer
440 views

Complex Fourier series

I need to find the complex Fourier series of this function, and I'm having problems calculating these integers: $$|a|<1$$ $$x\in [-\pi,\pi]$$ $$f(x)=\frac{1-a\cos(x)}{1-2a\cos(x)+a^2}$$ ...
0
votes
2answers
5k 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.
3
votes
5answers
495 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 ...
7
votes
6answers
3k 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 : $$\sum_{n=1}^\infty \frac{\cos nx}{n}$$ ...
2
votes
2answers
1k 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 ...
0
votes
1answer
331 views

Cesaro summable implies that $c_{n}/n$ goes to $0$

Theorem. If $\sum_{n=1}^{\infty}c_{n}$ is Cesaro summable, then $c_{n}/n$ tends to $0$. How to prove it?
1
vote
2answers
281 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 ...
3
votes
1answer
380 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: ${-\pi} \leq x \leq {\pi}$.
0
votes
1answer
175 views

Inverse Fourier transform to find out $\hat c_1$

If we have an integration which is need to solve inversely $$a_0 e^{-r^2/R^2} = \int_0^\infty \hat{c}_1(k) \frac{\sin(k r)}{r} dk,$$ If I transform the $\sin(kr)$, then we get imaginary part. Please ...
23
votes
4answers
37k 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 ...
10
votes
4answers
429 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}} ...
11
votes
3answers
786 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. ...
7
votes
2answers
4k 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 ...
6
votes
2answers
528 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): ...
5
votes
1answer
1k 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 ...
3
votes
2answers
246 views

If square waves are square integrable, why doesn't fourier expanding work?

If square waves are square integrable, then why does expanding on a fourier basis not recover the equation?
3
votes
1answer
337 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 ...
2
votes
2answers
182 views

Prove Parseval Identity for $f \in C(\Bbb T) 2\pi$ periodic continuous functions

Question: Prove Parseval Identity for $f \in C(\Bbb T) $ $2\pi$ periodic continuous functions $$ \frac{1}{2 \pi} \int_{-\pi}^\pi |f(x)|^2 dx =\sum_{n=-\infty}^\infty |\hat f(n)|^2 $$ Thoughts: We ...
4
votes
1answer
197 views

Parseval's Identity (Integral)

Calculate the integral: \begin{equation} \int_{-\pi}^{\pi}\left|\sum_{n=1}^{\infty}\frac{1}{2^{n}}e^{inx}\right|^{2}dx\end{equation} I'm familiar with Parseval's identity which states that for ...
2
votes
3answers
4k 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 ...
2
votes
1answer
545 views

Roots of a finite Fourier series?

In general, are there any clever tricks to help find the roots of a finite Fourier series? Presumably there aren't analytic methods, but can we use the fact that our function is a finite Fourier ...
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 ...
6
votes
1answer
767 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 ...
5
votes
1answer
569 views

fourier expansion of $\coth$ and justifying an identity

The problem: Justify the following equalities: $$\cot x = i\coth (ix) = i \sum^\infty_{n=-\infty} \frac{ix}{(ix)^2+(n\pi)^2}=\sum^\infty_{n=-\infty}\frac{x}{x^2+(n\pi)^2}$$ I am trying to figure ...
1
vote
4answers
24k 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 ...
6
votes
6answers
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 : $A_{0}+\sum_{n=1}^{\infty}(A_{n} \cos{nx} + B_{n} \sin{nx})$ ...
4
votes
1answer
252 views

Fourier series of $\sqrt{1 - k^2 \sin^2{t}}$

I'm struggling with a Fourier series. I need to find the Fourier series of the following function. That's the function under study: $f(t)=\left[\sqrt{1-k^2\sin^2t}\,\right]$. The function ...
3
votes
1answer
215 views

An integral that might be related to the modified Bessel function of second kind

It is known that the modified Bessel Function $K_z(a)$ ($a>0$)can be expressed as a Fourier transform $$K_z(a)=\frac{1}{2}\int_{-\infty}^{\infty}\exp(-a\cosh t)\cosh(zt){\rm d}t=K_{-z}(a)$$ Can ...
2
votes
1answer
227 views

Series expansion of $\coth x$ using the Fourier transform

Hi I have research about the series of coth but all of the solutions emerges from integral on a contour, Could you calculate the fourier transform of coth? Is that possible at all?My goal is to reach ...
2
votes
1answer
649 views

Absolute convergence of Fourier series of a Hölder continuous function

Suppose that $f$ is $2 \pi$ periodic and Hölder continuous of order $\alpha > 1/2$. Show that the Fourier series of $f$ converges absolutely. So we know that $f(x+2 \pi t) = f(x)$ for all $t \in ...
1
vote
3answers
2k 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 ...
0
votes
1answer
55 views

Riemann Lebesgue Lemma Clarification

If $f$ is continuous real-valued function, does the Riemann Lebesgue Lemma give us that $\int_{m}^k f(x) e^{-inx}\,dx \rightarrow 0\text{ as } n\rightarrow \infty$ for all $m\le k$? Specifically, is ...
6
votes
2answers
479 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
232 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 $j= 1 \dots n$. ...
3
votes
1answer
128 views

Convergence of series of functions: $f_n(x)=u_n\sin(nx)$

Let $f_n(x)=u_n\sin(nx)$ where $\displaystyle\sum f_n$ converges pointwise, and $ \displaystyle x \mapsto \sum_{n=0}^{+\infty} f_n(x)$ is continuous. Prove that $ u_n\rightarrow 0$ when n ...
3
votes
4answers
294 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
383 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 ...
2
votes
2answers
504 views

Please check my answer to $\sum_{i=1}^n \frac{\sin{(ix)}}{i} < 2\sqrt{\pi}$

$$\sum_{i=1}^n \frac{\sin{(ix)}}{i} < 2\sqrt{\pi}$$ I have this answer, please let me know if there is a more beautiful proof. My answer: at first, we prove two inequalities: If $x\in ...
1
vote
1answer
52 views

Finding the limit that involves Fourier coefficients,

Given the function $f(x) = 1 - \dfrac{|x|}{\pi}$, I had computed its Fourier coefficients, using integration by parts and got: $$ a_n = \begin{cases} 0, & \text{for $n$ even}, \\[6pt] ...
1
vote
1answer
33 views

Find the coefficients of the Fourier series that minimise the error.

I am having a little trouble understanding what I have to actually do here. What does differentiate with respect to bn? I thinks after differentiation I must use some calculus theorem about extreme ...
1
vote
3answers
714 views

Calculating own dft via matlab?

We are asked to code our own dft function from the formula : If everything is done correctly it should give the same result with matlab's own dft function, in the end I'm comparing them but they ...
1
vote
1answer
466 views

Function not satisfying pointwise convergence and Fourier series

Can you show an example of a function that does not satisfy pointwise convergence theorem hypotheses for Fourier series but that is still expressible as Fourier series? [Added after comment] In ...
1
vote
0answers
119 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 $f(x)$". ...
0
votes
0answers
34 views

How to find the Fourier coefficient of this function

How can we find the Fourier coefficient of the following function, f(x)=0 if -$\pi$$\leq$x<0 f(x)=1, if 0$\leq$x<$\pi$ and f(x)=f(x+2$\pi$) 1)how can we figure out that f(x) is integrable ...
0
votes
1answer
52 views

Prove that the function $\xi\in R \mapsto {e^{i\cdot \xi\cdot λ}-1\over i\cdot \xi}-λ$ is $C^{\infty}$ [closed]

Prove that the following function is $C^\infty$ in the point $\xi=0$: $$f:\Bbb R\to\Bbb C:\xi\mapsto {e^{i\cdot\xi\cdot λ}-1\over i\cdot\xi}-λ$$ Any ideas how to prove this? I am trying to ...
0
votes
1answer
110 views

Finding the Fourier series of a piecewise function

I'm s little confused about Fourier series of functions that are piecewise. Here’s an example of such a function: $$f(x) = \begin{cases} x & -\frac\pi2 < x < \frac\pi2 \\[5pt] \pi - x & ...
0
votes
1answer
205 views

How to use Parseval' s( Plancherel' s) identity?

Let $f\in L^{2} (\mathbb R)$ with $\lim_{t\to \pm \infty} f(t)=0.$ Put, $F_{n} (x)= \frac{1}{2\pi} \int_{-n}^{n}e^{itx} f(t) dt, \ (n=1,2,...).$ Fix $\alpha \in (0, \infty)$ and we define $H_{n}(x)$ ...