Questions tagged [fourier-series]
A Fourier series is a decomposition of a periodic function as a linear combination of sines and cosines, or complex exponentials.
3,501 questions
0
votes
0answers
21 views
Prove that the alternating sum of the odd reciprocals is $\frac{\pi}{4}$ using Fourier and Parseval's theorem [on hold]
Using the Fourier series of the function $f(x)=-k$ if $-\pi<x<0$ and $f(x)=k$ if $0<x<\pi$ and applying Parseval's theorem, prove that:
$$1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+...=\frac{\...
3
votes
1answer
23 views
Difference real and complex fourier series
I'm working on fourier series and I'm trying to compute the fourier transformation for the $2\pi$-periodic function of $f(x)=x^2$ with $x \in [-\pi,\pi]$.
Now with the real way, that is $$f(x) \sim \...
0
votes
0answers
31 views
Simpson vs. trapezoidal rule for numerically integrating $\cos{x}\cosh{x}$ in range 0 to $\pi$?
I have to numerically calculate many integrals similar to this:
$$\int_0^\pi \cosh{\left(\frac{a_1\cos{x}+a_2\cos{2x}+a_3\cos{3x}+\ldots}{10}\right)}\cos{jx}\cos{kx}\space dx$$
Right now I am using ...
1
vote
1answer
24 views
Find a trig polynomial, $(P_N)_N$, s.t. $(P_N)_N\rightarrow |x|$ on $\Big[-1/2,1/2\Big]$
Let $f$ be the $1$-periodic function such that $f(x) =|x|$ for $x \in \Big[ \frac{-1}{2},\frac{1}{2} \Big].$
Determine explicitly a sequence of trigonometric polynomials $(P_N)_N$ such
that $P_N \...
1
vote
0answers
14 views
Elliptic Fourier Descriptors - how to derive the coefficients?
In Kuhl and Giardina's 1982 paper, it derives the coefficients, $a_n$ and $b_n$, by equating the two definitions of $\dot{x}(t)$:
$$
\begin{align*}
\dot{x}(t) &= \sum_{n=1}^\infty \alpha_n \cos\...
1
vote
0answers
24 views
When Fourier coefficients are real?
Consider a periodic piecewise continuous complex valued function defined on the real line, under what conditions its Fourier coefficients are all real? Or, can they be complex?
0
votes
1answer
29 views
Fourier series coefficients for a piecewise periodic function
I have the following question:
The non-zero Fourier series coefficients of the below function will contain:
The answer is: $a_0, b_n, n=1, 3, 5, \cdot \cdot \cdot$
So I first tried to ...
0
votes
0answers
34 views
Find assessment of residual member and prove it's convergence [on hold]
I want to find a relationship between $N$ and $ \varepsilon $.
Evaluation of this expression is required:
$$ |R_N(x,t)|= |\sum_{n = N+1}^\infty \frac{4\pi\mu_n sin(\mu_n) sin(\frac{\mu_n x}{l}) \...
-3
votes
1answer
27 views
I was wondering whether or not these are equal to each other [on hold]
Does this $\frac{− cos (π n ) − 1}{πn}$ equal to $\frac{1}{nπ (1+(-1)^n+1)}$ ?
0
votes
1answer
26 views
analysis and fourier series [on hold]
Let $f$ a continuous fonction on $[-\pi,\pi]$, I would like to show :
$\forall n\in \mathbb{N},\quad \left.\begin{array}{ll}\displaystyle \int_{-\pi}^{\pi}f(t)\cos (nt)\cdot dt=0\\\\\displaystyle \...
0
votes
1answer
74 views
Find the Fourier transform of $e^{- x^2}$.
I am trying to get the cosine Fourier transform of $e^{- x^2}$ where I am stuck in simplifying the integration further. The integration is between the limits $0$ to $\infty$!
1
vote
1answer
30 views
Suppose that $\hat{f}(n) = -\hat{f}(-n)\geq 0$ holds for all $n \in \mathbb{Z}$. Prove that $\sum^{\infty}_{n=1} \frac{\hat{f} (n)}{n} < \infty.$
Let $f$ be $1$-periodic and continuous. Suppose that $\hat{f}(n) = -\hat{f}(-n)\geq 0$ holds for all $n \in \mathbb{Z}$, where the $\hat{f}(n)$ denotes the $nth$ Fourier coefficient of $f$. Prove that
...
2
votes
1answer
45 views
Where does this derivation of the Fourier Series for csc(x) go wrong?
In this post, the following derivation for the Fourier series of csc(x) is given:
\begin{align}
\csc x
&= \dfrac{1}{\sin x}\\
&= \dfrac{2i}{e^{ix}-e^{-ix}}\\
&= \dfrac{2ie^{-ix}}{1-e^{-...
8
votes
3answers
98 views
Showing that $-\frac{1}{2}=\sum^{\infty}_{n=1}\frac{(-1)^{n}\sin(n)}{n}$
So I've been trying to prove that
$$\sum^{\infty}_{n=1}\frac{(-1)^{n}\sin(n)}{n}=-\frac{1}{2}$$
I've tried putting various bounds on it to see if I can "squeeze" out the result. Say something like (...
1
vote
1answer
22 views
Derivation of the Dirichlet Kernel in Fourier Analysis.
$$D_N(x)=\sum^{N}_{-N}e^{2\pi inx}=e^{-2\pi iNx}\sum^{2N}_{0}e^{2\pi inx}$$
$$=e^{2\pi iNx}\left(\frac{e^{2\pi i(2N+1)x}-1}{e^{2\pi ix}-1}\right)$$
$$=\frac{e^{2\pi i(N+1/2)}-e^{-2\pi i(N+1/2)x}}{e^{\...
0
votes
0answers
19 views
How do lie symmetries manfest in the metric tensors of their manifolds?
Suppose we are considering some pseudo-Riemannian manifold (spacetime) that has an underlying topology $SU(2)\times U(1)$ (which incidentally corresponds to a closed cyclic universe). Such a space is ...
6
votes
3answers
87 views
Weird Integral Involving Hermite Polynomials
I have stumbled upon the following integral involving the Hermite polynomials:
$$ I(m) = \int_\mathbb{R} e^{i m x} \left[ e^{-\frac{x^2}{2}} H_m(x) \right] dx \, , \quad m \in \mathbb{N} \cup \{0\} \...
0
votes
0answers
14 views
Real Part of the Dilogarithm
It is well known that
$$\frac{x-\pi}{2}=-\sum_{k\geq 1}\frac{\sin{kx}}{k}\forall x\in(0,\tau),$$
which gives
$$\frac{x^2}{4}-\frac{\pi x}{2}+\frac{\pi^2}{6}=\sum_{k\geq 1}\frac{\cos(kx)}{k^2}.$$
...
0
votes
1answer
11 views
Upper bound of a partial sum of Fourier coefficients
This problem is from Stein & Shakarchi's Fourier Analysis, Exercise 16(b) of Chapter 3.
Let $f$ be a $2\pi$-periodic function which satisfies a Lipschitz condition with constant $K$; that is, $$|...
0
votes
1answer
20 views
Fourier Series of Product of Continuous Functions
Suppose $f, g$ are two continuous $1$-periodic functions. Prove that $$\widehat{f \cdot g}(n)=\sum_{m\in \mathbb{Z}}\hat{f}(n-m)\hat{g}(m),$$
where $\widehat{f\cdot g}$ is the Fourier coefficient of $...
2
votes
0answers
32 views
What is the Product of Fourier Coefficients?
Let $f, g$ be continuous $1$-periodic functions. Show that $$\widehat{fg}(n)=\sum_{m\in \mathbb{Z}}\hat{f}(n-m)\hat{g}(m).$$
Here the $\widehat{f\cdot g}$ denotes the Fourier coefficient of $f\cdot g$...
1
vote
0answers
26 views
Understanding a Fourier Series Problem
I found the following problem on Fourier Series. But I cannot make sense about how $f(x)$ could be a $1$-periodic function on the interval $[-1,1]$? Could you please help me make sense out of this ...
0
votes
1answer
17 views
Why a DFT of 2 senes is very noisy even with frequence sampling 5 times bigger?
I've set up this case to try to understand DFT implementing a real case in Excel
Frame Size $\;\color{blue}{(T}$): 5 s
Time Sampling $\;\color{blue}{(TS}$): 0,1 s
Block Size $\;\color{blue}{(N = TT/...
3
votes
1answer
138 views
Fourier-Bessel Series for $f(x)=1-x$
I am trying to find the Fourier-Bessel series for the function $$f(x)=1-x\ \ \text{for} \ \ 0<x<1.$$
The Fourier-Bessel series has the form $$f(x)=\sum_{n=1}^{\infty} A_nJ_v(k_nx), \ \ 0<x&...
1
vote
0answers
19 views
Does The Fourier Series of $\sin(px)$ uniformly converges when $p \notin \Bbb Z$
As a part of a question that I solving about Fourier Series, I had to decide if the Fourier Series in the domain $[-\pi,\pi]$of $\sin(px)$ is uniformly convergent.
Notice that $p>0$ and $p \notin \...
2
votes
1answer
55 views
Using Fourier Analysis to solve Basel Problem
Let $f$ be the $1$-periodic function such that $f(x)=x$ for $x \in [0, 1)$.
Compute the Fourier coefficient $\hat{f}(n)$ for every $n\in \mathbb{Z}$ and use Parseval’s theorem to derive the formula
$$\...
0
votes
0answers
33 views
Theta function equation
I have been trying to prove an equation of a theta function.
I understood that it some how related to the Poisson summation formula, but no luck. Any help would be appreciated.
the exercise
2
votes
1answer
14 views
Find coefficient of fourier series knowing the series for a similar function
Knowing that $f(\theta)=\pi - \theta$ defined on $[0,2\pi]$ has the fourier series $\sum \limits _{n=1}^{\infty}\frac{2\sin(n\theta)}{n}$, then find the coefficients of $g(\theta)=\pi-\theta$ defined ...
2
votes
0answers
67 views
changes the sign of a continuous $f(x)$ with period 2$\pi$ at least 2n+2 times on a interval $[a,b]$ with $b-a>2\pi$
Let $f(x)$ is continuous on $\mathbb{R}$ with period $2\pi$, if there exists $n \geq 1, n \in \mathbb{N}$ we have
\begin{equation}
\int_0^{2\pi} f(x) dx = \int_0^{2\pi} f(x) \sin xdx= \int_0^{2\pi} f(...
3
votes
1answer
46 views
What is the relation between separation of variables and the eigenfunctions and eigenvalues for PDEs?
Studying Fourier Series and its application of solutions for Partial Differential Equations, in particular (historically) for the heat equation, one starts by separating variables. Somehow related to ...
0
votes
1answer
66 views
If $f$ is $2\pi$-periodic and $C^{k}$, $\hat{f}(n)=O\left(\frac{1}{|n|^{k}}\right)$ as $n\rightarrow\infty$
My question is:
Suppose that $f$ is $2\pi$-periodic, and $C^{k}$. Show that $\hat{f}(n)=O\left(\frac{1}{|n|^{k}}\right)$ as $n\rightarrow\infty$. This notation means that there is a constant $C$ such ...
7
votes
2answers
159 views
“Bad” Fourier Series derivation
Let $f(\theta)$ $2\pi$-periodic such that $f(\theta)=e^{\theta}$ for $-\pi<0<\pi$, and $$e^{\theta}=\sum_{n=-\infty}^{\infty}c_{n}e^{in\theta}\,\,\, \mathrm{for}\,\, |\theta|<\pi $$
it's ...
5
votes
2answers
73 views
Proving that $\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^{2}}=\frac{\pi^{2}}{12}$
I want to prove that $$\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^{2}}=\frac{\pi^{2}}{12}, $$
using the Fourier Series for the $2\pi$-periodic function $f(\theta)=\theta^{2},\quad (-\pi<0<\pi)$, ...
0
votes
2answers
22 views
$L^1$-convergence
In my fourier analysis script we learnt that the fourier series of a function $f\in L^1[0,1]$ must not converge to $f$ in $L^1$, i.e. in general we have
$||S_n(f)-f||_{L^1[0,1]}\not\to 0$.
$S_n(f)$ ...
0
votes
2answers
47 views
Proving that $\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{4n^{2}-1}=\frac{\pi -2}{4}$
I need to prove the equation below:
$$\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{4n^{2}-1}=\frac{\pi -2}{4}$$
I think that I can use the following result provided by Fourier Series:
$$|\sin\theta|=\frac{...
1
vote
2answers
38 views
Fourier Series integral proof
Consider a Fourier series representation of a function $f(x).$ Let $S_{N}(x)$ be the $N^{\text{th}}$ partial sum defined by
$$S_{N}(x) = \frac{a_{0}}{2} + \sum_{n=1}^{N}a_{n}\cos\frac{n\pi x}{L} + ...
0
votes
1answer
23 views
proof that a fourier series converges to a pi periodic function
enter image description here
so given the information on the left, how did they on the right hand conclude the first equality that $$S_{N}(f;x)-f(x) = \frac{1}{2pi}\int_{-\pi}^{\pi}g(t)sin(N+\frac{1}{...
0
votes
2answers
41 views
Has the fourier series for all smooth periodic functions finitely many terms?
When a fourier series has only finitely many terms, $f(x) := \sum_{n=-N}^N f_n e^{inx}$, then it is obvious that $f$ is smooth and periodic. Is the converse true aswell?
If we have a smooth periodic ...
3
votes
1answer
33 views
Is mathematical relation between functions the same as relation between their Fourier series?
For example I have three functions $a(x)$,$b(x)$ and $c(x)$ which are defined on $(-\pi,\pi)$
They have a relation that $a(x)=2b(x)+c(x)$. Assume their Fourier series are $A(x)$ $B(x)$ and $C(x)$.
...
3
votes
1answer
128 views
Find the Fourier-Bessel Series for $f(x)$ With Respect to the Orthogonal Set: How Was $w(x)$ Found?
I have the following problem:
If $f(x) = x$, $0 < x < 2$, find the Fourier-Bessel series for $f(x)$ with respect to the orthogonal set $\{ J_1 (k_n x) \}$, where $k_n$ is the $n$th positive ...
2
votes
0answers
46 views
Looking for an explicit $f\in C(\mathbb{T})$ such that $t\mapsto\sum_{n=0}^{\infty}\hat{f}(n)e^{int}\notin L^\infty(\mathbb{T})$.
Denote the 1-torus with $\mathbb{T}$ and if $f\in L^1(\mathbb{T})$ denote the Fourier transform of $f$ by $\hat{f}$.
I know that
$$\exists f\in C(\mathbb{T}), t\mapsto\sum_{n=0}^{\infty}\hat{f}(n)e^{...
1
vote
2answers
25 views
Going from Fourier sum to Fourier integral - confusion on intermediate step
How does it solve the $A_0$ term?
Can one prove this with an example? I don't know how to prove it to myself with an example
In my lecture notes, my lecturer is trying to justify $C_n$ as weighting ...
0
votes
1answer
34 views
Fourier coefficients decay
This question popped up in a recent exam of mine to which I didn't know how to answer. The problem was as following
Given the function $$f(\theta) = \frac{1+\cos(\theta)}{3+2\cos(\theta)}$$ and it'...
3
votes
0answers
86 views
Convergence of Fourier series in Bergman norm
Let $D$ be the unit disk, let $\nu>-1$, let $\varphi_\nu(w):=(1-|w|^2)^\nu$, let $\operatorname{d}\mu_\nu:=\varphi_\nu\operatorname{d}\mu$ where $\mu$ is the Lebesgue measure on the unit disk. If $...
1
vote
1answer
38 views
If the series of the Fourier coefficients is absolute convergent then the Fourier series is uniform convergent
It the following correct?
If the series of the Fourier coefficients is absolute convergent then the Fourier series is uniform convergent.
Is it related to some theorem? Parseval's identity?
2
votes
0answers
18 views
Poincare type inequality for functions with vanishing fourier coefficients
Let $\mathbb{T}=\mathbb{R}/\mathbb{Z}$. Suppose $f\in L^1(\mathbb{T})$ and $f$ is also differentiable. Also assume that $f'\in L^p(\mathbb{T})$ for any $1\le p\le\infty$. Suppose that the Fourier ...
0
votes
1answer
36 views
Finding the value of $\sum_{n=1}^\infty \frac{1}{n^2}$ using Fourier transformation of a function f(x) [duplicate]
$f(x)=|x+\frac{\pi}{2}|-2|x|+|x-\frac{\pi}{2}|$
After using Fourier transformation I get
$a_0 = \frac{\pi}{2}$
$a_n = \frac{8}{\pi n^2}sin^2(\frac{n\pi}{4})$
$f(x)= \frac{a_0}{2} + \sum_{n=1} ^{\...
1
vote
0answers
77 views
Cremona 2.14.1 Why is $c_4$ and $c_6$ complex when they should be rational?
In Cremona's online book Chapter 2, in order to calculate the lattice invariant we have:
$\tau=\omega_1/\omega_2$
Set $q=e^{2\pi i\tau}$
(2.14.1) $c_4 =(2\pi/\omega_2)^4(1+240 \sum_{n=1}^{\infty} n^...
3
votes
0answers
23 views
Wavenumber $k$ becoming a continuous variable as $\lambda \to \infty$
I'm trying to piece together how to get to derive the Fourier transform from the Fourier series, reading my lecturer's notes.
Now, if $f(x)$ happens to not be periodic, we
can still give a ...
1
vote
0answers
32 views
Fourier Series Of $a^2-x^2$
Find Fourier coefficients of $
f(x)=
\begin{cases}
a^2-x^2, \mid x\mid<1 \\
0, \mid x\mid\geq1\\
\end{cases}
$ when $x\in (-\pi,\pi)$
What is the value at $x=1$
for which values of $a$ the series ...
