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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1answer
18 views

Complex Fourier Series of $t^3$

I am trying to find compute the complex Fourier series of the following function: $$f(t) = t^3$$ $$-\frac32 \le t \le \frac32$$ $$f(t) = f(t+3)$$ I am using the generic function for the complex ...
0
votes
1answer
38 views

The Fourier transform of exp(-x)*heaviside(x)

I'm trying to understand the Fourier transform of Y=exp^-x. Since the term tends to -infinity I have to multiply Y by the heaviside function to set everything below 0 to 0 so I can successfully ...
-1
votes
0answers
26 views

find the fourier transform of $xf(x)$ appended

I've seen the method in which you prove this fourier transform, but what if you don't recognize that $$xf(x) e^{i k x} = \frac{1}{i} \frac{\partial}{\partial k} \Big[ f(x) e^{i k x} \Big] $$ would I ...
1
vote
1answer
11 views

Show behavior of Fourier Transform

If F(w) is the Fourier transform of f(x), show that F(aw) is the Fourier transform of (1/a)f(x/a). So if I apply a fourier transform to (1/a)f(x/a): $$ \frac{1}{2\pi}\int_{-\infty}^\infty ...
2
votes
0answers
21 views

I would like to find a generalization of the plane wave expansion to Hankel functions.

The plane wave expansion is \begin{equation} e^{i\vec{k}\cdot \vec{x}}=\sum_{\ell=0}^{\infty}i^\ell(2\ell+1)j_{\ell}(kx)P_{\ell}(\cos(\theta)) \end{equation} where $j_\ell$ is the spherical Bessel ...
0
votes
1answer
15 views

Find the Fourier Transform of piecewise finction

$$f(x) = \begin{cases} 0 & |x|> a \\ 1 & |x|< a \end{cases}$$ I have most of the solution, I'm just faltering on obtaining the sin(ax) part of the solution, I'm missing an exponential ...
1
vote
3answers
26 views

How do I find the solution to this summation after computing the following power series?

I have found that the Fourier cosine series from $({-\pi},{\pi})$ of the function $f(x)=\cosh(x)$ is $$ \frac{2\sinh({\pi})}{\pi}\left[\frac{1}{2}+ \sum_{n\: =\: 1}^{\infty}\:\ ...
4
votes
2answers
51 views

Integral using Parseval's Theorem

How would I integrate $$\int_{-\infty}^{+\infty} \frac{\sin^{2}(x)}{x^{2}}\,dx$$ using Fourier Transform methods, i.e. using Parseval's Theorem ? How would I then use that to calculate: ...
0
votes
1answer
47 views

Solve differential equation using fourier series

I am trying to solve this problem in my analysis book in a chapter on Fourier series: Solve the differential equation $$(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}) u(x,y) = ...
0
votes
0answers
14 views

Chladni patterns

So I was watching this video on Chladni figures (https://www.youtube.com/watch?v=wvJAgrUBF4w) and thought that it would be nice to replicate a few of these, especially the more complicated, high ...
0
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0answers
23 views

Table of Fourier series

I found that there are very good references on Fourier integral transform but none on Fourier series. Do you happen to know one?
1
vote
1answer
14 views

ODE with finite Fourier expansion periodic coefficients

Regard the ordinary differential equation $$ \dot a(t) = z(t) a(t) $$ where $a(t)$ and $z(t)$ are matrix valued such that $z$ is periodic ($z(t+2\pi)=z(t)$). Then it is well-known (Floquet theory), ...
0
votes
0answers
11 views

Compute the Fourier series of a piecewise function.

Consider the function: $ f(\theta) = \begin{cases} 0 & \text{if } |\theta| >\delta \\ 1-|\theta|/\delta & \text{if } |\theta| \leq \delta \end{cases} $ I need to show ...
0
votes
0answers
34 views

Show that the Fourier series is $\frac{8}{\pi} \sum_{k \;odd \ge 1} \frac{sin(k \theta)}{k^3} $

Consider the odd function $f(\theta)=\theta (\pi - \theta)$, then I need to show that: $f(\theta)=\frac{8}{\pi} \sum_{k \;odd \ge 1} \frac{sin(k \theta)}{k^3}$ then I computed the Fourier ...
2
votes
2answers
37 views

Show that $\widehat{f}(n)$ is zero for odd $n$

The following problem is from Stein´s Introduction to Fourier analysis: Suppose that $f(\theta + \pi)=f(\theta)$ for all $\theta \in \mathbb{R}$ Show that $\widehat{f}(n)$ is zero for odd $n$. My ...
1
vote
1answer
28 views

Writing a Fourier series of a $2\pi$-periodic function.

This problem was taken from Stein's Introduction to Fourier analysis, and it goes like this: Let $f$ be a $2\pi$-periodic Riemman integrable function defined on $\mathbb{R}$. Show that the Fourier ...
0
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0answers
13 views

Fourier Series from a simple sinosoidal function

I have a function of: Vd = (Rd/(33e3+Rd))*(Vrf*sin(Wrf*t)+Vlo*sin(Wlo*t)); y_t = 11*Vd; func_ = y_t; where: ...
2
votes
2answers
55 views

A difficult trigonometric integral involving absolute value

$$ \int_{0}^{2\pi}\lvert\sin(x)\rvert\cos(nx)\,dx= -\frac{4\cos^2\bigl(\frac{\pi n}{2}\bigr)\cos(\pi n)}{n^2-1} $$ I'm not actually trying to solve this myself. The answer appears in my lecture notes ...
2
votes
1answer
68 views

A trigonometric integral identity from Krylov's “Approximate Calculation of Integrals”

In the theory of Fourier series the following expansion is known $$ \operatorname{sign}\left(\sin\left((n + 1) x\right)\right) = \frac{4}{\pi} \sum_{k = 0}^\infty \frac{\sin\left((2k + 1) (n + 1) ...
0
votes
1answer
24 views

Do I have to transform the solution into $u(x, y)$?

Find the solution of the problem $$u_{xx}(x,y)+u_{yy}(x,y)=0, x^2+y^2>1 \\u=1+3\sin^3 \theta , 0 \leq \theta <2\pi$$ $u$ is bounded. I have done the following: $$u(x,y)=v(\rho, \theta) \\ ...
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votes
0answers
17 views

Can a modular form every have a polynomial expansion in q?

In other words, if q = e^(2*piiz), where z is in the upper half plane, can the fourier expansion of a non-constant modular form ever have a finite number of terms?
0
votes
1answer
30 views

Can we expect $\|fg\|_{\mathcal{F}L^{1}} \leq C \|f\|_{L^{2}(\mathbb R)} \|g\|_{\mathcal{F}L^{1}}$?

Consider the space of all Fourier transforms of $L^{1}(\mathbb R),$ that is, $$\mathcal{F}L^{1}=\mathcal{F}L^{1}(\mathbb R):= \{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}(\mathbb R)\},$$ with the ...
3
votes
1answer
38 views

$\|fg\|_{A (\mathbb T)} \leq C \|f\|_{L^{2}} \|g\|_{A (\mathbb T)}$?

Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$.Consider the space, $$A(\mathbb T):= \{f\in ...
0
votes
1answer
127 views

Is there a closed-form of $\sum_{n=1}^{\infty} \frac{\sin(n)}{n^4}$

Is there a closed-form summation result for Fourier series: $$\sum_{n=1}^{\infty}\frac{\sin(n)}{n^4}?\tag{1}$$ I tried using available result of the following (odd) function : ...
2
votes
1answer
30 views

On the weak closedness of a closed ball with fixed $L^2$-norm in a periodic Sobolev space

Preliminaries: Let $\mathrm{L}_P^2$ denote the Hilbert space of $P$-periodic, locally square-integrable functions $f\colon \mathbb{R} \to \mathbb{C}$ with Fourier series representation $$f(x) \sim ...
1
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0answers
20 views

Properties of Fourier coefficients of real valued functions

Let $\hat{f}(n)$ be the Fourier coefficients of $f:[0,2\pi]\to \mathbb{C}$ defined as $$\hat{f}(n)=\int_{0}^{2\pi}f(x)e^{-{\rm{i}}nx}\,\mathrm{d}x$$ Note $f$ is Riemann-integrable on $[0,2\pi]$. We ...
0
votes
0answers
9 views

Finding Fourier coefficients of (discrete ) $cos(\frac{6*n*\pi}{N})$

What is the Fourier coefficients of (discrete ) $cos(6*n*pi/N)$? The answer says $0.5[delta(k-3)+delta(k+3)]$ (delta is Dirac delta function)...my attempt was to use a formula $1/N(sum from 0 to ...
0
votes
0answers
6 views

Solutions of $\sum_{n=1}^N a_n n\sin{(n x+\theta_n)}=\sum_{n=1}^N a_n n^2\cos{(n x+\theta_n)}=0$

Is there a solution for the equation $\sum_{n=1}^N a_n n\sin{(n x+\theta_n)}=\sum_{n=1}^N a_n n^2\cos{(n x+\theta_n)}=0$ in terms of the variable $x$, for some choice of coefficients $a_n$ and ...
1
vote
1answer
33 views

periodicity of an exponential sum

I wish to rigorously prove that the function $f(x), x \in \mathbb{R}$ is not periodic. A function is defined to be periodic with period $M$ if $f(x+M)=f(x), \forall x \in \mathbb{R}$. Here $f(x) ...
0
votes
0answers
18 views

Saddle points, local maxima and minima and Fourier sum

Assume I have function defined as a Fourier sum in the form $f(x)=\sum_{n=1}^N a_n \cos{(n x+\theta_n)}$ where I assume that $a_n\neq0$ and that $N>1$. and that I am interested on determine the ...
0
votes
0answers
36 views

Green's function of Harmonic Oscillator using Fourier modes

First off, I know this is similar to an already answered question concerning the Green's function of a harmonic oscillator. I wanted to ask a question there in the comments, but couldn't due to ...
1
vote
1answer
21 views

Fourier series-odd and even functions

f+ is the even part of the function and f- is the odd part. I'm not able to understand how it is that they got the values of modulus of x and x for the even and odd parts of the function ...
2
votes
4answers
146 views

A Fourier Analysis Question I am stuck at

If $f,g\in C[-\pi,\pi]$,and $f,g$ are $2\pi$ periodic, prove that $$\lim_{n\to\infty}\dfrac{1}{2\pi}\int_{-\pi}^\pi f(t)g(nt)\mathbb dt=\big(\dfrac{1}{2\pi}\int_{-\pi}^\pi f(t)\mathbb ...
1
vote
0answers
23 views

How can I make the mean of samples be approximately equal to the mean of actual continuous signal?

Suppose there is signal f(t) that is continuous and periodic. It is known that this f is T-periodic. (but it's not necessarily a single cosine f(t).( I'd like to make the mean of samples be ...
0
votes
2answers
33 views

I don't understand the relation.

$$e^ix - 1 =e^{ix/2}* 2i * sin({x/2}) $$ I don't understand why that is true, but I do know the relation $$sin (x) = \frac {e^{ix} - e^{-ix}}{2i}$$ However I don't see where the 1 came from
0
votes
1answer
16 views

Frequency scaling property for Fourier series

For Fourier transform, there is an equation connecting time-scaling with frequency-scaling. (By scaling, I mean multiplying by constant for time or frequency) Is there such a relation for Fourier ...
3
votes
5answers
82 views

Can anyone suggest a book on Fourier Analysis containing many good problems

I am taking a basic course in Fourier Analysis in my undergrad Analysis class and I know the theory and related theorems. However, this is a relatively new zone for me and I would like a book that ...
0
votes
0answers
8 views

Fourier Amplitude Sensitivity Test (FAST)

I am new in the domain of sensitivity analysis, I am trying to investigate the global sensitivity analysis method FAST (Fourier Amplitude SEnsitivity testing). I read alot about this subject, starting ...
4
votes
2answers
72 views

Solving a PDE by Fourier Series

I want to solve the following PDE: $$\begin{cases} u_t=u_{xx}+1\\ u_x(0,t)=0, \quad u(1,t)=0\\ u(x,0)=\cos\left(\frac{\pi}{2}x\right) \end{cases}$$ using a Fourier series. The thing that is throwing ...
0
votes
1answer
38 views

Find the Fourier series representation of $f(t)=\sin(3\pi t)$

Find the Fourier series representation of $$f(x)=\sin(3\pi t)\qquad \text{for }-1\leq t\leq1$$ When I calculate the coefficients, I always get $0$. Why is that? Is the series indeed zero?
2
votes
1answer
65 views

Number of zeros of a periodic function

Let's consider a periodic real function of a real variable $f(x)$. If the function is analytical and it is not the zero function, can one infer that the number of zeros in one period $[x,x+P)$ is ...
0
votes
0answers
20 views

Is there anything similar to DTFT for Fourier series?

So if sampling condition is met well, with aperiodic signals we have discrete-time Fourier transform (DTFT) that allows us to get frequency-domain data that resemble continuous-time fourier transform. ...
2
votes
2answers
41 views

Fourier integral problem?

Show that $$ \int_0^{\infty} \frac{\sin \pi \omega \sin x\omega}{1-\omega^2}d\omega= \begin{cases} \frac{\pi}{2}\sin x,&\mbox{ if } 0\leq x\leq\pi\\ \quad\\ 0,&\mbox{ if } x\geq\pi ...
0
votes
1answer
34 views

What is a window function with positive spectrum?

I need a real, symmetric window function $x(t) = x(-t)$ whose Fourier transform $\hat{x}(\omega)$ (also real and symmetric) is non-negative $\hat{x}(\omega) \ge 0$ for all $\omega$. The function does ...
2
votes
1answer
22 views

Is the DTFT of a sampled Gaussian a positive function?

I have an infinite sequence $x_{n}$ for $n \in \mathcal{Z}$ which is a sampled Gaussian function $x_{n} = \exp(-n^2/a)$ with a > 0. I need to check whether its DTFT $x(\theta) = \sum_{n \in ...
1
vote
1answer
31 views

Find the Fourier Coefficients that minimize the error [duplicate]

I know that the coefficients that minimize the expression are the ones that make it's derivative 0. I have also expanded the whole expression and taken it's derivative, but still I can't figure out ...
0
votes
1answer
33 views

differentiation and integration of Fourier series.

If I have the fourier series of $|x|$ for $-l < x < l$ and I make it periodic with period $2l$ I get a cos series: $$ \frac{l}{2} ...
1
vote
0answers
9 views

Result obtained on deletion of finite number of Fourier Coefficients

I want to know the answer to the following question. If a finite (but fixed) number of Fourier coefficients (of any choice) of a Fourier series are made $0$, then will the new series be a Fourier ...
0
votes
2answers
22 views

DTFT and its convergence

In the textbook "signals and systems", by prof. Simon Haykin, it says:   If $x[n]$ is not absolutely summable, but does satisfy square summable, then it can be shown that the following equation ...
0
votes
2answers
26 views

Find complex Fourier coefficients of $f(-x), f^*(x)$

For $f(-x)$ i have tried to replace the $k$ with $k'=-k$ but still i can't find any relationship between the coefficients. What could be a better way to approach this problem?