# 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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### Homogeneous solution, Fourier mode, wavelength

Consider a system $$\partial_t u=N(\sigma)u,~~u=(x,t)~~~(1)$$ where $N$ is a non-linear operator depending on some control parameter. Suppose that the system (1) admits a homogeneous ...
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### Averaging transformation of a closed plane curve

Let's suppose we have a closed plane curve of some shape whose points are described by the single parametric equation $P(x(t), y(t))$ where $t$ is some increasing parameter (example circle) or by ...
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### Equidistributed problem about polynomial with irrational coefficient

This problem is from Stein, Fourier Analysis,Chapter 4,problem 2(d). Problem:Suppose that $P(x)=c_n x^n+……+c_0$ is a polynomial with real coefficients, where at least one of $c_1,……,c_n$ is ...
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### Calculating a uniform-in-wavelength Fourier transform

If $f$ is frequency in Hertz and we assume $c = 1$, then $\lambda = 1/f$ and we have the representation$$s(t) = \int_{-\infty}^\infty \hat{s}(1/\lambda)\exp(i2\pi t/\lambda)1/\lambda^2\,d\lambda.$$How ...
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### Relationship between Fourier coefficients, eigenvalues, and the spectrum of a ring for dummies?

As the question title suggests, what is an explanation for dummies of the relationship between Fourier coefficients, eigenvalues, and the spectrum of a ring?
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Theorem: If $u \in L^2(\mathbb{R}^n)$ then the Fourier transform $\widehat{u} \in S'(\mathbb{R}^n)$ is a $L^2(\mathbb{R}^n)$ function and the Parseval's identity holds: $||\widehat{u}||_{L^2(\mathbb{R}... 1answer 45 views ### Can you recover a distribution from mollification? Let$f\in \mathcal S'(\mathbb R)$be a Schwartz distribution. Given$\rho \in C^\infty_c(\mathbb R)$define the convolution as the function $$x\mapsto (f\ast\rho)(x):=\langle f, \rho (\cdot -x)\... 2answers 40 views ### Finding the coefficients of a triangular wave. I have the following equation that I want to solve$$a_k = \color{blue}{\frac{1}{T} \int_{0}^{T/2} 2 \frac{t}{T} e^{-i \frac{2\pi}{T}kt} dt} + \color{red}{\frac{1}{T} \int_{T/2}^{T} 2 \frac{T-t}{T} e^... 2answers 54 views ### Relation between Dirac function and inverse fourier transform of 1 According to my notes, it holds that$\delta=(2 \pi)^{-n} \widehat{1}$. How do we get the equality? We have that$\delta=\frac{1}{(2 \pi)^n} \int_{\mathbb{R}^n} \widehat{\delta}(\xi) e^{i x \xi} d{\...
i have the following exercice: Let for all $x \in \mathbb{R},$ $f(x)= \cos x$ and $g(x)= \sin x$. Calculate $T=f \delta' + g \delta''$ for this question, i find $T=3 \delta$. Calculate the Fourier ...
My ultimate goal is to show that the Hilbert transform of a Schwarz function is in $L^p(\mathbb{R})$, for every $p \in (1,\infty]$ (the definition I am using is \$Hf(\xi) := \mathcal{F}^{-1}[(-i \ \...