Tagged Questions

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

Here's the function (I need it's fourier transform).
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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Spectrum of a Rectangular signal?

I would be grateful if you can help me with i) ii) and iii) especially with ii and iii
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Fourier Analysis and applications to Abels groups

Find all functions $f:A\rightarrow C$ such that: $$\sum_{x\in A}|(f*f)(x)|^2 = |A|(\sum_{x\in A}|f(x)|^2)^2$$
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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. $$(1+\cos(x-1))^3$$ $$= ... 1answer 40 views f(x) =\cos(x-y) -\cos(\delta) plotting Ok, so this is a confusing one. I'm not sure what my teacher is looking for. The problem is: Plot any number -\pi < y< \pi and pick a small number \delta > 0 so that the whole interval ... 1answer 35 views Fourier analysis question Let f(t)=\frac 12 -t, t\in(0,1). Calculate the Fourier coefficients of the function f and the sum \sum_{n=1}^{\infty} \frac {1}{n^2}. Note that L^2 (\Bbb{T}) \to l^2(\Bbb{Z}) and ... 1answer 41 views DFT matlab problem We have signal \sin(2\pi v_1 t)+\sin(2\pi v_2 t) and we know ν_1\in{700,780,860,940} and ν_2\in{1200,1340,1480}. Also we have vector here:$$h(k)=\sin(2π ν_1 k Δt)+\sin(2π ν_2 k Δt)$$where ... 0answers 38 views fourier matlab question Let's assume that s(-3)=3, s(1)=3, s(5)=-1 s(-2)=4, s(2)=1, s(6)=-4, s(-1)=4, s(3)=2, s(0)=-3, s(4)=-3 and s(k+10)=s(k). for this signal the discrete fourier transform ... 1answer 51 views proof by induction to fourier problem So if h_n (t) = e^{\pi t^2}\frac{d^n}{dt^n}(e^{-2\pi t^2}). Show proof by induction that$$\widehat{h_n}=(-i)^n h_n$$Any ideas how to go about with this one? When n=0 \to \widehat{h_0}=h_0. 1answer 43 views Fourier conjugate problem? Do I need to use conjugation rules to show that \overline{h(t)}=\hat{g}(t) and when g(t)=\overline{\hat{h}(t)}? Trying to prove parsevals identity with this one. Edit: Something like this: ... 0answers 37 views Convolution + fourier So let's say we have s(t)=e^{-a|t|} so I calculated that \hat{s}=\frac{2a}{a^2 + 4 \pi^2 v^2}. Then if h_a = \frac{2a}{a^2 + 4\pi^2 t^2}. using \hat{s}, how can we prove that h_a * h_b = ... 1answer 60 views Laplace question How do you express Laplace transform \mathcal{L}(g)(z)=\int_{0}^\infty e^{-zt}g(t)dt with Fourier transform? And how do you form the reverse formula for Laplace transform using Laplace transform ... 1answer 73 views Fourier transform of a cosine function I was reviewing a homework problem, and I'm trying to figure this out. The Fourier transform of {1\over 2} cos(3\pi t), according to the solution I was given is {1\over 2}\{\delta(f+{2\over ... 1answer 44 views Why is \frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\cos\left[ a\xi \right]\hat{f}(\xi)d \xi = f(a)? Background: We are looking at the wave equation on \mathbb{R}^n via the Fourier transform. If u(x,t) solves \Delta u = u_{tt} in \mathbb{R}^n, with u(x,t) = f(x) at t=0 and u_t(x,t) = ... 1answer 910 views Fourier series coefficients proof Can somebody help me understanding the fouries series coefficients? I know that if we have:$$f(n) = \sum_{n=1}^N A_n \sin(2\pi nt + Ph_n) \tag{where $Ph_n$ = phase} And because of the ...
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 ...