# Tagged Questions

Harmonic analysis is a generalisation of Fourier analysis that studies the properties of functions. Check out this tag for abstract harmonic analysis (on abelian locally compact groups), or Euclidean harmonic analysis (eg, Littlewood-Paley theory, singular integrals). It also covers harmonic ...

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### The Fourier Transform of taking Eigenvalues

The purpose of this question is to ask about the Fourier transform of the map which associate to an $n$ by $n$ matrix its $n$ eigenvalues, or some function of the $n$ eigenvalues. The main motivation ...
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Let $f\in L^{1}(\mathbb R)$ and it Fourier transform, $\hat{f} (y) : = \int _ {\mathbb R} f(x) e^{-2\pi i x\cdot y} dx ; y \in \mathbb R ;$ and consider Fourier algebra $$A(\mathbb R):= \{f\in L^{1}(... 0answers 256 views ### Scattering for rapidly decaying solutions of NLS Cazenave and Weissler proved in their paper "Rapidly Decaying Solutions of the Nonlinear Schrödinger Equation" the following property. Given the problem \left\{ \begin{array}{rl} ... 0answers 266 views ### What information about a locally compact group G is encoded in C_r^\ast(G) which is not in L^1(G)? Let G be a locally compact group and let  C_r^\ast(G)  denote its reduced group C^\ast-algebra. Many features of a G can be realized from L^1(G) or C_r^\ast(G). For example, G is ... 0answers 172 views ### Fourier analysis on crystallographic groups It seems pretty clear a priori that Fourier analysis on a discrete cocompact group of motions of \mathbb{E}^n should be quite tractable (since, by Bieberbach, such groups are almost \mathbb{Z}^n, ... 0answers 242 views ### Smoothness of the convolution of a singular measure with itself Let \gamma:I\subset\mathbb{R}\rightarrow\mathbb{R}^2, s\mapsto \gamma(s), denote the arclength parametrization of a smooth, convex curve \Gamma:=\gamma(I)\subset\mathbb{R}^2. Equip \Gamma with ... 0answers 392 views ### Measure Theoretic view of Hardy Littlewood Circle Method Is it possible to view the Hardy-Littlewood Circle method as the Fourier transform with respect to the Lebesgue measure on [0,1) for an appropriate generating function defined in terms of additive ... 0answers 86 views ### Optimal Kakeya Maximal Bound for Bushes Let \{T_{\alpha}\} be a collection of 1\times\cdots\times 1\times N tubes, where N\gg 1, with maximal 1/N-separated directions, which all are centered at the origin (i.e. they form a bush). In ... 0answers 76 views ### interpretation of a singular integral There is a post on MSE about a principal value integral in this paper. It has not received much attention even with a bounty, and since it concerns a published paper, I believe this is a better forum ... 0answers 102 views ### Ideals of L^1(G) I want to study the closed ideal structure of L^1(G). Is there a good paper or book which characterizes closed ideals and maximal ideals of L^1(G)? 0answers 163 views ### Non-compact analogue of Peter-Weyl I have the following situation: G is a real unimodular locally compact semisimple Lie group. Then it is known that the regular representation H:=L^2(G,\mu_H) decomposes as \int^{\... 0answers 66 views +50 ### Global Euclidean Carleman Estimate with a linear phase I am interested in deriving the following global Carleman estimate which I think should hold :  \| e^{\tau \phi} \triangle e^{-\tau \phi} u \|_{L_{\delta+1}^2({\mathbb{R^3})}}> C \tau \| u \|_{L^... 0answers 66 views ### Almost sure convergence of double nonconventional ergodic averages with respect to L^p function A famous result of J. Bourgain says that for a probability measure preserving system (X,\beta,\mu,T), with T_1 and T_2 powers of T, we have that for f_1, f_2\in L^{\infty}(\mu),$$\frac{1}{...
Suppose $f\in L^p(\mathbb{R}^d)$. Then the Dirichlet BVP for the Laplace equation $\Delta u = 0$ in the upper-half plane $\mathbb{R}^d\times\mathbb{R}_{>0}$ with boundary value $f$ can be solved by ...
Let $p(x) \geq 0$ be a positive polynomial on the hypersphere ($x \in S^{n-1}$) satisfying $\int_{S^{n-1}} p(x) = 1$. Writing $p(x) = \sum_{j=0}^s p_j(x)$ where $p_j(x) = \sum_m p_{jm} s_{jm}(x)$ with ...