1
vote
0answers
44 views

Fourier transform help for solving $u_t+u_{xxxx}+u_{xx}=0$

I just started to learn a little bit of fourier analysis in solving PDEs. I want to find a solution $u(x,t)$ to $u_t+u_{xxxx}+u_{xx}=0$. My attempt: Applying the fourier transform to both sides gives ...
0
votes
0answers
49 views

Fourier transform of a function

I'm struggling with FT, I just can't grasp the concept of it. Can somebody explain it on an example Ex 1: $f(t) = e^{-|t|}$ EX 2: $x(t) = \cos(\pi t/T)$ where it's different from $0$ just on ...
-2
votes
2answers
40 views

Fourier transform of t*(sent/pi*t)^2

Here's the function (I need it's fourier transform).
1
vote
1answer
55 views

Fourier series problems

I've got an "interesting" problem. I've gotten a way through it, but I'd like someone to look if what I've done so far is correct, and what to do next. We've got a function that is $0$ on the ...
2
votes
1answer
36 views

Show that the Fourier transform of a a distribution is $C^{\infty}$

I am trying to understand the solution to the following problem: Let $u \in \mathcal{D}'(\mathbb{R}^{n})$ such that $u(x) = c \log(|x|)$ when $|x|>1$, where $c \in \mathbb{C}$. Show that $u \in ...
0
votes
1answer
40 views

Fourier and $Z$ transform of a signal?

We have $$X(k)=4[u(k-2)-u(k)* d(k-3)]$$ I need to find the Fourier transform,$Z$ transform,as well as dhe magnitude and phase spectra. First of all I think that I need to convert the $u(k)$ and ...
1
vote
0answers
46 views

Show that the Hilbert transform is a pseudo-differential operator of order 0

I have tried to solve the following exercise but I don't know if I have missed anything. Show that the Hilbert Transform $Hf = \mathcal{F}^{-1}(\operatorname{sgn}(\xi)\hat{f}(\xi))$ is a ...
1
vote
1answer
56 views

Find distribution solving a differential equation

I think I have solved the following differential equation, but I am not sure of all steps are justified. Exercise: Find all distributions $u \in \mathcal{D}'(\mathbb{R})$ such that $x(u' -u) = ...
0
votes
1answer
57 views

Find limit of a sequence of distributions

I am trying to solve the following exercise: Determine the limit in $\mathcal{D}'(\mathbb{R})$ of $\lim_{t\rightarrow \infty} t^{2}xe^{itx}$, $u_{t} = t^{2}xe^{itx}$. I have tried evaluating the ...
3
votes
1answer
96 views

Prove $\sum_{n= -\infty}^{\infty} \frac{1}{(t+n)^2} = \frac{\pi^2 }{\sin^2(\pi t)}$ using f(x)=1-|x| and Poisson summation formula

I'd like to prove $$\sum_{n= -\infty}^{\infty} \frac{1}{(t+n)^2} = \frac{\pi^2 }{\sin^2(\pi t)}$$ by using the Poisson summation formula. There is a way to do it by firstly taking the Fourier ...
0
votes
1answer
31 views

Show that $f(x)$ is orthogonal to $f'(x)$ in $L^2(-\pi, \pi)$

I have the following problem: Suppose $f$ is of class $C^{(1)}$, $\;2\pi$-periodic, and real-valued. Show that $f'$is orthogonal to $f$ in $L^2(-\pi, \pi)$ by a) expanding $f$ in ...
1
vote
3answers
82 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
50 views

Exists $C$ constant: $||f||_{L^{\infty}(\mathbb{R})}\leq\{ ||f||_{L^2(\mathbb{R}}+||f'||_{L^2(\mathbb{R})} \}, \forall f\in S(\mathbb{R})$

Show that there exists $C$ constatant such that $||f||_{L^{\infty}(\mathbb{R})}\leq\{ ||f||_{L^2(\mathbb{R}}+||f'||_{L^2(\mathbb{R})} \}, \forall f\in S(\mathbb{R})$. This is a question in my ...
0
votes
1answer
18 views

Understanding a Fourier analysis problem

I have the following problem from my Fourier analysis book: where $PC(a,b)$ is the set of piecewise-continuous functions. I don't quite understand my task in this problem, what am I supposed to ...
0
votes
2answers
25 views

Fourier Transform of $f(t+a)$ if $f(t)$ has tranform $F(k)$?

I know the formula $$f(t) = \int^{+\infty}_{-\infty} F(k)e^{ikt} \, dk$$ and I've seen that for computing $f'(t)$ it's a case of differentiating $e^{ikt}$ inside the integral, so $f'(t)=ikF(k)$ Can ...
1
vote
0answers
60 views

Using Fourier Transforms to Solve the Heat Equation PDE In Infinite Three Dimensions

Problem: Using Fourier transforms, solve for $u(x,y,z,t)$, where $$u_t=D\nabla^2 u$$ $$ -\infty<x,y,z<\infty,t>0$$ $$D>0, u(x,y,z,0)=f(x)f(y)f(z)$$ and $u\rightarrow 0$ as ...
2
votes
0answers
29 views

Good family of kernels in $\mathbb{R}^n$

I'm trying to prove that, given the heat equation $u_t = \Delta u$ with boundary values $u(x,0) = f(x)$, the solution given by $$u(x,t) = f \star H_t^{(d)}(x)$$ is continuous up to the boundary ...
0
votes
1answer
47 views

Multiply a circulant matrix by a vector with FFT.

I am asked to write a Matlab program to find the coefficients of the resulting polynomial which is the product of two other polynomials. However, I need someone to clarify the underlying concepts for ...
2
votes
0answers
21 views

d-dimensional integral of exponential and determinant

I'm working on a question from Stein and Shakarchi's Fourier Analysis. This is exercise 5 from chapter 6: Let $A$ be a $d \times d$ positive definite symmetric matrix with real coefficients. Show ...
0
votes
1answer
27 views

Spectrum of a Rectangular signal?

I would be grateful if you can help me with i) ii) and iii) especially with ii and iii
1
vote
2answers
46 views

fourier series analysis, show that for every integer n, using euler's formulas relating trigonometric and exponential functions

Show that for every integer $n$, $$\int_0^{\pi} \cos nt~\sin t~\mathrm{d}t = \begin{cases} \dfrac{2}{1-n^2} & \text{if } n \text{ is even} \\[10pt] 0 &\text{if } n \text{ is odd} ...
1
vote
1answer
44 views

Hausdorff-Young inequality

Let $1<p\leq2\leq q \leq \infty$ and let: $$ \frac{1}{p} + \frac{1}{q}=1 $$ prove that for all finite Abel groups and all functions $f:\mathbb{A}\rightarrow \mathbb{C}$ Hausdorff-Young ...
0
votes
0answers
22 views

Polynomial Factorization - 5 variables

Factorize a polynomial with 5 variables $P\in C[x_0,x_1,x_2,x_3,x_4]$, $$\sum^{4}_{j=0}x_j^5 + ...
0
votes
0answers
18 views

Fourier Analysis and applications to Abels groups

Find all functions $f:A\rightarrow C$ such that: $$\sum_{x\in A}|(f*f)(x)|^2 = |A|(\sum_{x\in A}|f(x)|^2)^2$$
0
votes
0answers
28 views

Reconstruction formula for a function of moderate decrease

I want to show that, for a function $f$ of moderate decrease and $\hat{f}(\xi)$ supported in $I=[-\frac{1}{2},\frac{1}{2}]$, $$f(x) = \sum^{\infty}_{n=-\infty} f(n)K(x-n)$$ where $K(y) ...
3
votes
1answer
67 views

Poisson summation formula clarification regarding Fejer kernel

Define $$\mathbf{F}_R(t) = \begin{cases} R \left(\dfrac{\sin(\pi R t)}{\pi R t}\right)^2 & t \neq 0\\[10pt] R & t = 0 \end{cases} $$ A problem in Stein's Fourier Analysis asks ...
2
votes
4answers
141 views

Functions that are their own Fourier transformation

In Stein's Fourier Analysis, there's an exercise: The function $e^{-\pi x^2}$ is its own Fourier transform. Generate other functions [presumably in the Schwartz space $S(\mathbb{R})$] that, up ...
1
vote
1answer
59 views

Simplified version of the Fourier inversion formula

I'm working out of Stein's Fourier Analysis (working with Riemann integrable functions), and I'm having trouble with problem 1: Suppose $f$ is continuous and supported on $[-M,M] \subset ...
0
votes
1answer
17 views

Fourier transform in space

I have an expression on the form $$ g_i(x+r\delta t, t+\delta t) = T_{ij}g_j(x,t) $$ and I would like for find its Fourier transform. According to my book it should be $$ g_i(k,t+\delta t) = ...
0
votes
0answers
13 views

Calculating the DFT of a sequence following a mathematical expression

This is homework, so please don't give a full solution. Give a formula for $F_k$ for all $k$ where $f_n=4^n$ for all $n=0,\dots ,N-1$. I ended up abusing Wolfram Alpha and getting probably way ...
1
vote
1answer
46 views

The behavior of Fourier transform near the origin

I'm attacking a homework problem, which I have reduced to the following: Let Schwartz function $f \in \mathcal{S}^1(\mathbb{R})$ be nonnegative and $\|f\|_{L^1} = 1$. Assume further that ...
3
votes
3answers
417 views

Fourier Series for $|\cos(x)|$

I'm having trouble figuring out the Fourier series of $|\cos(x)|$ from $-\pi$ to $\pi$. I understand its an even function, so all the $b_n$s are $0$ $$a_0 = \frac 2 \pi \int_0^\pi |\cos(x)|\,dx = ...
2
votes
1answer
43 views

Equidistribution of $\{\xi_n\}$ where $\xi_n = <n\frac{p}{q}>$ $p,q$ rel. prime

I'm working from Stein's An Introduction to Fourier Analysis, and there's a question (chapter 4 number 6): Let $\theta = \frac{p}{q} \in \mathbb{Q}$ where $\operatorname{gcd}(p,q) = 1$. Assume ...
0
votes
1answer
37 views

A Specific Example about Parabolic PDE

I am solving a PDE numerical problem. And I have already had a algorithm. However, it seems to be hard to find a specific example to test my solution. Could you please give me one? The equation ...
0
votes
1answer
42 views

Fourier analysis how do i calculate an equation

Struggle is an understatement! I'm trying to get my head around Fourier analysis and I have the equation : $$f(x)=2\pi^2+6x^2$$ unfortunately I have no idea where to start and my coursework depends ...
1
vote
1answer
78 views

Laplace transform of $g_n(t)=\begin{cases}\frac{(1-e^{-t})^n}{t^n}&:t>0,\\0&:t\le0.\end{cases}$

Find Laplace transform for this function "$g$" $$g_n(t)=\begin{cases}\frac{(1-e^{-t})^n}{t^n}&:t>0,\\0&:t\le0.\end{cases}$$ Then Take advantage of it to calculate the following ...
0
votes
0answers
19 views

Determine whether the set is uniqueness set

We say that $\Lambda$ is a uniqueness set for the Paley-Wiener space $PW_{\pi}$ if $$(F \in PW_{\pi} \wedge F|_{\Lambda}\equiv 0) \rightarrow F\equiv 0.$$ For example, $\Lambda =\mathbb Z$ is a ...
3
votes
1answer
65 views

Express $(1+\cos(x-1))^3$ as a trigonometric polynomial in x.

Express $(1+\cos(x-1))^3$ as a trigonometric polynomial in x. I keep doing this problem and somehow I keep messing up the constants, and it just jumbles up in my head. $$(1+\cos(x-1))^3$$ $$= ...
0
votes
1answer
40 views

$f(x) =\cos(x-y) -\cos(\delta)$ plotting

Ok, so this is a confusing one. I'm not sure what my teacher is looking for. The problem is: Plot any number $-\pi < y< \pi$ and pick a small number $\delta > 0$ so that the whole interval ...
1
vote
1answer
35 views

Fourier analysis question

Let $f(t)=\frac 12 -t, t\in(0,1).$ Calculate the Fourier coefficients of the function $f$ and the sum $\sum_{n=1}^{\infty} \frac {1}{n^2}$. Note that $L^2 (\Bbb{T}) \to l^2(\Bbb{Z})$ and ...
1
vote
1answer
41 views

DFT matlab problem

We have signal $\sin(2\pi v_1 t)+\sin(2\pi v_2 t)$ and we know $ν_1\in{700,780,860,940}$ and $ν_2\in{1200,1340,1480}$. Also we have vector here: $$h(k)=\sin(2π ν_1 k Δt)+\sin(2π ν_2 k Δt)$$ where ...
0
votes
0answers
38 views

fourier matlab question

Let's assume that $s(-3)=3$, $s(1)=3$, $s(5)=-1$ $s(-2)=4$, $s(2)=1$, $s(6)=-4$, $s(-1)=4$, $s(3)=2$, $s(0)=-3$, $s(4)=-3$ and $s(k+10)=s(k)$. for this signal the discrete fourier transform ...
2
votes
1answer
51 views

proof by induction to fourier problem

So if $h_n (t) = e^{\pi t^2}\frac{d^n}{dt^n}(e^{-2\pi t^2})$. Show proof by induction that $$\widehat{h_n}=(-i)^n h_n$$ Any ideas how to go about with this one? When $n=0 \to \widehat{h_0}=h_0$.
0
votes
1answer
43 views

Fourier conjugate problem?

Do I need to use conjugation rules to show that $\overline{h(t)}=\hat{g}(t)$ and when $g(t)=\overline{\hat{h}(t)}$? Trying to prove parsevals identity with this one. Edit: Something like this: ...
0
votes
0answers
37 views

Convolution + fourier

So let's say we have $s(t)=e^{-a|t|}$ so I calculated that $\hat{s}=\frac{2a}{a^2 + 4 \pi^2 v^2}$. Then if $h_a = \frac{2a}{a^2 + 4\pi^2 t^2}$. using $\hat{s}$, how can we prove that $h_a * h_b = ...
1
vote
1answer
60 views

Laplace question

How do you express Laplace transform $\mathcal{L}(g)(z)=\int_{0}^\infty e^{-zt}g(t)dt$ with Fourier transform? And how do you form the reverse formula for Laplace transform using Laplace transform ...
1
vote
1answer
73 views

Fourier transform of a cosine function

I was reviewing a homework problem, and I'm trying to figure this out. The Fourier transform of ${1\over 2} cos(3\pi t)$, according to the solution I was given is ${1\over 2}\{\delta(f+{2\over ...
0
votes
1answer
44 views

Why is $\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\cos\left[ a\xi \right]\hat{f}(\xi)d \xi = f(a)$?

Background: We are looking at the wave equation on $\mathbb{R}^n$ via the Fourier transform. If $u(x,t)$ solves $\Delta u = u_{tt}$ in $\mathbb{R}^n$, with $u(x,t) = f(x)$ at $t=0$ and $u_t(x,t) = ...
1
vote
1answer
910 views

Fourier series coefficients proof

Can somebody help me understanding the fouries series coefficients? I know that if we have: $$f(n) = \sum_{n=1}^N A_n \sin(2\pi nt + Ph_n) \tag{where $Ph_n$ = phase}$$ And because of the ...
4
votes
1answer
75 views

Easy question about fourier transform

I think this is an easy one ... but I just can't find an answer. Assume $f : \mathbb{R} \to \mathbb{R}$ is a $n+2$ times differentiable function and and all drivatives up to the order $n+2$ are in ...