1
vote
0answers
107 views
+50
Fourier analysis questions
Can anyone give me a hand with the proof of this properties?
Prove that:
a) The linear span of the set $\left\{T_bh/b\in\mathbb{R}\right\}$ is dense in $L_2(\mathbb{R})$, where $h(x)=e^{-\pi x^2}$. ...
1
vote
1answer
37 views
Fourier analysis question, orthonormal basis.
I need some help with this exercise:
Given $A>0$, let $L_{A}^2(\mathbb{R})$ the subspace of $L^2(\mathbb{R})$ of the functions $f$ that satisfy $\hat{f}=\chi_{[\frac{-A}{2},\frac{A}{2}]}\hat{f}$. ...
1
vote
1answer
33 views
Parseval's identity
How to prove the Parseval's identity , I know the formal way but how to justify the interchange between the integral and the sum in a rigorously way , in addition what extra condition does the ...
2
votes
1answer
68 views
In my Fourier text book, there are the following exercises to prove. why do some of them have the same left side but have different right sides?
In my Fourier text book, there are the following exercises to prove.why do some of them have the same left side but have different right sides? The demand of these question is to prove these ...
3
votes
2answers
157 views
A Fourier series exercise
Can anyone give me a hand with this exercise about Fourier series?
Let $f(x)=-\log|2\sin(\frac{x}{2})|\,\,\,$ $0\lt|x|\leq\pi$
1) Prove that f is integrable in $[-\pi,\pi]$.
2) Calculate the ...
9
votes
4answers
277 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 $f$,
$$
\int_{-\infty}^{\infty} f(x) e^{-Ax^2}\,dx = \sqrt{\frac{\pi}{A}} ...
1
vote
0answers
45 views
Intervals where the function is similar to the Fourier series
$$f(x)=\left\{\begin{array}{l l}
0,\quad x \in [-L,0[\\
1,\quad x \in [0,L]
\end{array}\right.$$
I need to know in which intervals the sum of the Fourier series is "equal to the function $f(x)$".
...
-1
votes
1answer
38 views
How to prove this Fourier question?
How to prove this Fourier question? I hope for a procedure in detail.
1
vote
0answers
63 views
Fourier Analysis of Prime Counting Function
I was thinking about the following:
Denote $\pi(x)$ as the prime counting function such that:
$$
\pi(x) = \#\text{ of prime numbers}\leq x
$$
It is well known from the prime number theorem that
$$
...
1
vote
0answers
22 views
Understanding the indices in a Fourier series
Sometimes the truncated Fourier series of a function with Fourier coefficients $\hat{u}_k$ is written
$$\sum_{k=-N}^N\hat{u}_ke^{ikx}$$
which is a linear combination of $\cos(nx) +i\sin(nx)$ for ...
0
votes
1answer
30 views
Show that Fourier coefficients approach zero uniformly
Let $f(t)$, $g(t)$ be piecewise continuous functons on $[-\pi,\pi]$, periodically continued on $\mathbb R$. I want to show that
$$
a_n(x) = \frac{1}{\pi} \int\limits_{-\pi}^{\pi} f(x+t)g(t) ...
-1
votes
0answers
41 views
Laurent expansion of this funtion
What is the Laurent expansion of this.
$$f(x)=\frac{2−a(z+z^{-1})}{2(1−az)(1−az^{-1})}=\frac{a}{2(z−a)}+\frac{1}{2(1−az)}+\frac{1}{2}$$
Please help me on this so I can calculate next the fourier ...
-2
votes
2answers
292 views
Calculate the Fourier transform of $b(x)=1/(x^2+a^2)$
I need help to calculate the Fourier transform of this funcion
$$b(x)=\frac{1}{x^{2}+a^{2}}$$
where $$a>0$$
Thanks
0
votes
1answer
32 views
Wave-Function Series?
So I was basically exploring the function:
$\displaystyle {\text{frac}(x)}$ which is the fractional part function and I noticed that it has a nice fourier series definition which is:
...
0
votes
1answer
54 views
Fourier transforms - don't understand this concept!!! Please help me on this
I have two Fourier transforms to solve, but the problem is that a I have a characteristic bijection or some etching that I don't know what it is and I don't know how to solve this... Please help
...
1
vote
1answer
51 views
Sum over cosines = dirac delta - how to get the coefficients?
Given this formula:
$$\sum\limits_{n=0}^\infty a_n \cos(n \pi x / d) = \delta(x-x_0)$$
Where $0 \leq x \leq d$. How can one calculate the coeffciients $a_n$?
I googled and searched all kinds of ...
1
vote
0answers
26 views
Complex Fourier series of a function [duplicate]
I need to find the complex Fourier series of this function, and I'm having problems calculating these integers:
$$|a|<1$$
$$x\in [-\pi,\pi]$$
$$f(x)=\frac{1-a\cos(x)}{1-2a\cos(x)+a^2}$$
...
2
votes
2answers
185 views
Complex Fourier series
I need to find the complex Fourier series of this function, and I'm having problems calculating these integers:
$$|a|<1$$
$$x\in [-\pi,\pi]$$
$$f(x)=\frac{1-a\cos(x)}{1-2a\cos(x)+a^2}$$
...
3
votes
2answers
120 views
Fourier Series of $f(x) = x$
I am having trouble finding the complex Fourier series of $f(x) = x$ and using that complex series to find 1)the real Fourier series of $f(x)$ and 2) the complex and real Fourier series of $h(x) = ...
0
votes
1answer
36 views
A function whose derivatives always have a convergent fourier series
I am looking for a solid example that such a function that its derivatives can always be found by taking derivatives component-wisely in its Fourier series. A function with finitely many Fourier terms ...
2
votes
2answers
64 views
How to solve this equation by Fourier series?
$$ y''+3y=\sin ^4 x ,\quad y=\frac{1}{8} +\frac{\cos2x}{2}-\frac{\cos4x}{104}.$$
Now the text book states the solution, but I don't know the process of solving this equation. I need your help!
1
vote
0answers
26 views
Upper bound on truncation error of a fourier series approximation of a pdf?
Given a probability density function, $f\left(x\right)$, of a continuous random variable, $X$, and given an $N$-th order fourier series approximation:
$$f_N\left(x\right)=\sum_{n=-N}^{N}c_n e^{inx}$$
...
2
votes
1answer
105 views
Pointwise convergence of double Fourier series
I'm looking for theorems that deal with the pointwise convergence of double Fourier series expansions for a special class of functions.
Let $D \subset [-\pi, +\pi]^2$ be an arbitrary set of finite ...
0
votes
1answer
43 views
Exponential Form of Fourier Series
Problem Suppose $f$ is a continuous function on interval $[-\pi,\pi]$ such that $\sum_{n\in\mathbb{Z}} |c_n| < \infty$ where $c_n = \dfrac {1}{2\pi} \int_{-\pi}^\pi f(x)\cdot \exp(-inx)~dx$, the ...
1
vote
4answers
106 views
Fourier Analysis
I am interested in Fourier Analysis. But I don't get why the coefficients are choosen that way and why the Fourier series will converge to a given function?
Can someone provide me simple information ...
0
votes
1answer
33 views
Function as a convolution product of other two
I need help with this:
I have to prove that a function $f\in L_{2}(T)$ can be expressed as $f=g*h$ (convolution product) for some functions $g,h\in L_{2}(T)$ if and only if $(\hat{f}(n))_{n}\in ...
1
vote
2answers
67 views
Fourier coefficients of the product of two functions
Given two functions $f,g\in L^2(\mathbb{T})$, I have to prove that the Fourier coefficients of $fg$ are given by
$$\hat{fg}(n)=\sum_{k\in{Z}}\hat{f}(n-k)\hat{g}(k)$$
and that this series converges ...
0
votes
0answers
34 views
Convergence of Fourier series - strange graph in proof
I am reading a text that states the following related to convergence of Fourier series:
$$g_K(x) =
> ...
3
votes
0answers
125 views
Compare Fourier and Laplace transform
I would like to clarify main difference between Fourier and Laplace transforms and also understand if exponential factor is main difference between this two method. So Fourier transform is ...
2
votes
3answers
119 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 ...
0
votes
1answer
82 views
Fourier Series of Multivariable Functions.
If I have some function $V(x,y)$ which is periodic in x with period L. I wish to expand $V(x,y)$ in terms of a fourier sine (for simplicity) series in $x$, is it always the case that I may write the ...
0
votes
0answers
37 views
Fouries series of $\sin{\sum_n a_n \sin{n\theta}}$
What's the Fourier series for $\sin({\sum_n a_n \sin{n\theta}})$?
0
votes
0answers
20 views
Probability distribution of a Fourier series
I have a real-valued signal which is represented by a finite Fourier series
$$
x(t) = \sum_{n=-N}^{N} \hat{x}_n e^{i \omega_n t}
$$
For this signal I would like to determine the probability $p$ of ...
2
votes
1answer
60 views
Fourier coeficients, convergence of the integral
Let $g\in L_{\infty}(\mathbb{T})$. For any $-\pi<a<b<\pi$ let $\chi_{[a,b]}$ be the characristic function of $[a,b]$. Prove that
...
3
votes
2answers
74 views
Identity involving partial sums of Fourier series
Suppose $f$ is a continuous periodic function and $S_Nf(x) = \sum^N_{n=−N} \hat f(n) e^{inx}$, where $$\hat f(n)= \frac{1}{2\pi}\int_0^{2\pi}f(x)e^{-inx} dx.$$
How can I show that ...
1
vote
0answers
19 views
Discrete time fourier transform of partial sum
I came across the following property of the DTFT:
$ \mathcal{F} \Bigg(\sum_{m=- \infty}^{n}x[m]\Bigg) = \frac{1}{1- e^{-j \omega}} X(e^{-j \omega}) + \pi X(e^{-j0}) \sum_{m= ...
3
votes
0answers
199 views
Solve a differential equation using Fourier series
Assume I have a second order differential equation $\ddot{x} = F(x,\dot{x})$ (or an equivalent equation of first order) and that I know there is a periodic solution to it (for simplicity's sake, ...
1
vote
0answers
59 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
43 views
Approximation using a Fourier transform with low pass filter
I need to approximate a function f, but I cannot do so with frequencies that exceed 1kHz
What is the best approximation I can get? Is taking the Fourier transform then zeroing any term above 1kHz the ...
2
votes
0answers
36 views
What are the connections between spectral expansion and differential operator?
For instance, for a nice function $f$ on the unit circle, we have its Fourier expansion,
$$f(x)=\sum_n \hat{f}(n) e^{inx},$$
where the exponentials are eigenfunctions for differential operator ...
1
vote
1answer
69 views
$\int_{-\infty}^{\infty}f(\xi)d\xi = \lim_{\delta \to 0 }\sum_{n=-\infty}^{\infty}\delta f(\delta n)$?
Assume that $f$ is continuous and moderate decrease, show that
$$\int_{-\infty}^{\infty}f(\xi)d\xi = \lim_{\delta \to 0,~ \delta>0}\sum_{n=-\infty}^{\infty}\delta f(\delta n)$$
From the ...
5
votes
0answers
275 views
Show that the function is constant
Let $S^n$ be an $n$-dimentional unit sphere.
Consider $f: S^n \longrightarrow R_+$ even continuous function.
Denote
$$
...
1
vote
0answers
59 views
Prove complex Fourier Series in 2D
Prove the complex form of Fourier Series in 2Dimension from periodic function (period $2\pi$) in $x$ and $y$, defined in region $\Omega\subset\mathbb{R^2}$
$$f(x,y)\sim\sum_{-\infty}^{\infty} ...
1
vote
0answers
31 views
Relations between complex functions satisfying a specific condition
What is the relation between the following two complex functions:
$$g(\theta)=\sum_n x[n]\ y[n]\ e^{in\theta}$$
and
$$f(\theta)=\sum_n \left(x[n]\pm i\sqrt{1-x[n]^2}\right)\ y[n]\ e^{in\theta}$$
...
0
votes
1answer
75 views
Help with some Fourier series questions
I need some help with this question about Fourier Series.
1) If $f\in{L_{1}(T)}$ (that's $f$ periodic with period $2\pi$ and $|f|\in{L_{1}([-\pi,\pi]}$))
with Fourier series ...
1
vote
1answer
106 views
FFT Algorithm for an interpolating polynomial
I'm trying to use the Fast Fourier transform algorithm to determine the trigonometric interpolating polynomial of degree $16$ for $f(x) = x^2\cos(x)$ on $[-\pi,\pi]$
I see a computer result in my ...
4
votes
2answers
88 views
How to visualise Fourier Transform of a function?
I solved many problems on Fourier series,transforms and inverse fourier transforms as part of my academics. And i am aware that FT converts a time domain signal to frequency domain and IFT is vice ...
3
votes
2answers
40 views
Fourier sums in cosine and sine and Borel resummation
is there a method to evaluate the fourier sums ??
$$ \sum_{n=0}^\infty t^n \sin(nx)= F(x,t) $$
$$ \sum_{n=0}^\infty t^n \cos(nx)= G(x,t) $$
my idea is that i need to use these sums to apply Borel ...
1
vote
1answer
34 views
Why does $\sin{\alpha}\cdot i\sin{\alpha x}$ disappear from this integral?
In a section on fourier transforms, my textbook contains these steps for an example:
$$f(x) = \int_{-\infty}^\infty \frac{\sin{\alpha}}{\pi \alpha}e^{i\alpha x}d\alpha$$
$$= ...
1
vote
1answer
58 views
Do we have a general form for this integral?
Is there a general formula or recursion for this integral?
$$\int_0^1\left(\frac{\arcsin x}{x}\right)^n\text{d}x,\ \ n\in\mathbb{N}$$
