# Tagged Questions

Fourier analysis, also known as spectral analysis, encompasses all sorts of Fourier expansions, including Fourier series, Fourier transform and the discrete Fourier transform (and relatives). The non-commutative analog is (representation-theory).

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### Fourier-Transformation of Operator

I have an operator $\hat{L}$ which gives $$\hat{L} f(x) = \lambda \cdot f(x)$$ where $\lambda$ is the eigenvalue. Now I Fourier-Transform my function $f(x)$: $$\mathcal{F}(f)(p) = g(p)$$ Question: ...
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### Complex Integration: theorem of Residues

I am trying to prove a theorem that is doing my head in a bit. I have tried to simplify the problem as much as possible and leave out the details, even though it might look a bit too big. The ...
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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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Let $f(x_1,x_2)=\frac{x_1}{x_2^{1/3}} \times \chi_{\{ [0,1] \times[-1,1]\}}$. Part A: Evaluate $\int_{\mathbb{R}^2} f(x_1,x_2) ~ \mathrm{d}x_1~ \mathrm{d}x_2$ $\int_{\mathbb{R}^2} f(x_1,x_2)~ \... 1answer 113 views ### Fourier series without Fourier analysis techniques It is known that one can sometimes derive certain Fourier series without alluding to the methods of Fourier analysis. It is often done using complex analysis. There is a way of deriving the formula $$\... 1answer 43 views ### Fourier Transform of exp(-a|x-.5|) So I've been working on the fourier transform of exp(-a|x-\frac{1}{2}|) (with a>0) and keep getting: \left(e^{-\pi i}\right)\left(\frac{2a}{a^+4\pi^2x^2}\right). A research partner keeps ... 1answer 57 views ### Levy processes, vanilla option and Fourier Transform The context to this problem is mathematical finance, although the answer does not need specific knowledge of the area. I am trying to work out the expression for the price of a call option using Levy ... 2answers 739 views ### Image Reconstruction:Phase vs. Magnitude Figure 1.(c) shows the Test image reconstructed from MAGNITUDE spectrum only. We can say that the intensity values of LOW frequency pixels are comparatively more than HIGH frequency pixels.$$ f(x,y)=... 1answer 108 views ### What are Basis images? I have read that using Fourier transformation we can decompose any arbitrary image into orthogonal basis images and reconstruct it back. But what are basis images actually? 0answers 94 views ### Explicit solution of the nonlinear Schrödinger equation Consider the linear Schrödinger equation, $$(LS) \begin{cases} \partial_{t}u= i\Delta u, t\in \mathbb R,\\ u(x,0)=u_{0}(x), \end{cases}$$$x\in \mathbb R^{n}.$Taking the Fourier transform with ... 1answer 54 views ### Does the integral in the formal 2D Fourier transform of the logarithm converge? If$k$is a nonzero vector in$\mathbb R^2$, how to interpret this integral: $$\int_{\mathbb R^2}e^{ik\cdot x}\ln{|x|}dx$$ Does it converge and in what sense? Thanks in advance. 1answer 204 views ### Fourier transform of$te^{-t^2}$? How can I find the Fourier transform of: $$f(t) = te^{-t^2}$$ 0answers 113 views ### Why are divergent Fourier series all so 'HARD'? I'm not sure if this question is appropriate or even making sense, but I still feel curious: why are every example of divergent Fourier series SO COMPLICATED? It usually takes pages to construct and ... 1answer 92 views ### A function sequence converge to a Fourier series implies point-wise converge? Assume$f(x)$is a smooth$2\pi$periodic function and can be decomposed as $$f(x)=\sum_{n=-\infty}^\infty a_ne^{inx}$$ A function sequence$f_m(x)$satisfies $$\int_0^{2\pi}f_m(x)e^{-inx}dx\to a_n,\... 0answers 83 views ### Building up the Fourier inversion theorem on locally compact abelian groups I'm reading through Folland's Abstract Harmonic Analysis and I've come to a bit of a road block with some machinery developed for the Fourier inversion theorem. We know that the Fourier transform of a ... 1answer 599 views ### Proof of Fourier series Theorem (k-continuous derivatives) Here's the theorem: Theorem: If f is periodic with Fourier coefficients a_n,b_n and if the series$$\sum_{n=1}^\infty (|n^{k}a_n|+|n^{k}b_n|)$$converges for some integer k \geq 1, then f ... 1answer 79 views ### Pseudodifferential operators and amplitudes I am studying psudodifferential operators on \mathbb{R}^n. Let U\subset \mathbb{R}^n an open subset. A function is b\in C^\infty(U\times U\times U \times \mathbb{R}^n) is an amplitude of order ... 1answer 169 views ### Characteristic function of logarithm of random variable If I know the characteristic function \phi_X(t) of a random variable X>0, how can I write the characteristic function \phi_Y(t) of Y=\log(X)? I know that \phi_X(t)=E[e^{itX}] and \phi_Y(... 1answer 47 views ### Discrete Fourier Transform I am studying DFT and am having trouble with the notation system. The frequency is from 0 to 2B - in DFT the frequency domain does not have negative frequencies. But if this is the case, and we ... 2answers 38 views ### \int\exp(-jnw_0t)\,dt integral calculus. I seem to forgot these parts of integral calculus. I am trying to determine the Fourier coefficient in complex exponential form. Here, t is the variable being integrated and n is for all ... 1answer 36 views ### How to calculate the value of \int_{-\infty}^{\infty} y(t)dt? For a function g(t), \int_{-\infty}^{\infty} g(t)e^{-j\omega t}dt=\omega e^{-2\omega ^2} for any real value \omega. If y(t)=\int_{-\infty}^{t}g(\tau)d\tau then how to calculate the value of \... 1answer 543 views ### Fourier Transform and its Inverse Could anyone show me how to prove the following results about Fourier Transform, please? It is stated in my book without proof. Thank you. Let \mathcal F denote the Fourier linear operator and f ... 1answer 50 views ### N-point FFT and 2-radix FFT I am wondering what is the difference between a N-point FFT (output has same length as the input) and a 2-radix FFT (output is always of length 2^n) For example a is a sequence: ... 1answer 70 views ### Is \langle f,g\rangle defined for distributions f,g? Consider a standard setting for the development of the theory of distributions. Let D(\Omega) be the space of test functions and D'(\Omega) be the space of distributions ("generalized functions"). ... 0answers 161 views ### How to do this Sum? Poisson Resummation? In the paper hep-th/0812.2909 page 34-35, there's a sum that I've been trying to do explicitly but I can't find a way. The sum is$$ \frac{2l}{\pi l! (l-1)!} \sum_{k\in\mathbb{Z}} \sum_{n=0}^{\infty} ... 2answers 113 views ### What does the Fourier transform of$1/x^2$mean? If I ask Mathematica to compute the Fourier transform of$\frac{1}{x^2}$using the FourierTransform function, it gives me a result of $$\mathcal{F}\biggl[\frac{1}{... 1answer 45 views ### intuition behind an identity related to fourier transforms I saw the proof of this identity in a question about Fourier transforms : F(f(−t))w=F(f(t))(−w) Can someone give the intuition behind it ? What I understand of Fourier transform of a function ... 0answers 42 views ### Proving something is a convolution operator… If we define the operator K(a)=F^{−1}aF where F:L^2({\mathbb R})\to L^2({\mathbb R}), is the fourier transform given by$$\left(Ff\right)\left(x\right)=\int_{{\mathbb R}}{f\left(t\right)e^{... 1answer 78 views ### Fourier transform of exp(cos) How do I calculate the Fourier transform ($t \rightarrow \omega$) of the following:$\exp(A\cos(\omega_0 t))A$is a real constant, and$\omega_0$is a real and positive constant. I know that this ... 1answer 37 views ### a question about Fourier transforms I know it s simple but how to show that$\mathcal F(f(-t))w=\mathcal F(f(t))(-w)$?$\mathcal F(f(-t))w=\int^\infty_{-\infty}f(-t)e^{-iwt}dtif -t=x\to -dt=dx\int^\infty_{-\infty}f(x).e^{iwx}(-...
I have to solve an exercise, but if i could use the following theorem, it would be piece of cake Similarity Theorem if $\mathscr{F}\{g(x,y)\}= G( f_x,f_y)$ then \$ \mathscr{F}\{g(ax,by)\}= \frac {...