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      Questions tagged [harmonic-analysis]

      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 analysis on tube domains, as well as the study of eigenvalues and eigenvectors of the Laplacian on domains, manifolds and graphs.

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      Reverse Loomis-Whitney Inequality for funcctions

      I was wondering if the reverse Loomis-Whitney inequality holds for general functions: Let $n\geq 2$. Let $(X_i,\mu_i)$, $1\leq i\leq n$ be measure spaces. Write $x=(x_1,\dots,x_n)$ and for each $1\...
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      Asymptotics of a function from its Fourier transform

      My question is: given a Fourier transform $\hat f$ of a function $f$, is it possible to estimate its asymptotic behaviour without performing the inverse transform? Let me give a concrete example. ...
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      A question about integration of spherical harmonics on $(S ^ 2, can)$

      Question: suppose that $H_{n_1}, H_{n_2}, H_{n_3} \in L^{2}(\mathbb{S}^2)$ are Spherical Harmonics of degrees $n_j$ $(j = 1, 2, 3)$ with $n_1 > n_2 + n_ 3$. Then, it is true that $$ \int_{\mathbb{...
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      135 views

      More general form of Fourier inversion formula

      My question begins as follows: Suppose $G$ is a compact (Lie) group and $A$ is a $G$-module, and denote the action of $G$ on $A$ by $\alpha$. Fix $a\in A$ and view $$ f:g\mapsto \alpha_g(a) $$ as an $...
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      30 views

      Characterizing geometrically Schwartz Kernels of pseudodifferential operators on a compact manifold

      Let $M$ be a compact smooth manifold without boundary. Define $\mathcal{P} \subset \mathcal{D}^{'}(M \times M)$ to be the smallest linear subspace of the space of distributions on the product which is:...
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      82 views

      Interchanging Integration Order involving Fourier Transform

      $$f(\omega,u):=\frac1{\omega+iu}$$ where $i$ is the imaginary unit number. We see that the integral of a Fourier transform $$\int_1^\infty du\int_{-\infty}^\infty d\omega\,f(\omega,u)\,e^{-i\omega x}=...
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      106 views

      Hardy-Littlewood in Sobolev Spaces

      For an application in kinetic theory I have to apply the Hardy-Littlewood-Sobolev(H-L-S) for $q=\infty$. The dimension is $3$ and H-L-S inequality says that for $1<p<q<\infty$ and $0<\nu&...
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      171 views

      Shattering with sinusoids

      Let $d \geq 2$ and $K$ some positive integer. Consider distinct points $\theta_1, \ldots, \theta_K\in \mathbb{T}^d$ and (not necessarily distinct) $z_1, \ldots, z_K \in \{-1,1\}$ such that $\sum\...
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      162 views

      Can one realize this as an ergodic process?

      Consider the lattice $\mathbb Z^2$ and take iid random variables $Y_e$ on all edges $e$ of the graph. We then define random variables $X_i:=\sum_{e \text{ adjacent to } i}Y_e.$ In other words: For ...
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      Failure of Schur's lemma for topological group representations

      Is there an example of $G$, $\rho$ as below? $G$ is a locally compact group. $\rho$ is an irreducible continuous representation of $G$ on a complex Hilbert space $V$. This means that we have a ...
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      201 views

      Ergodic theorem and products

      If $f_n (\omega) = \sum_{i=1}^n f_1 (T^i \omega)$ and $T$ is an ergodic action with respect to the measure $\mu$ then it is know as Birkhoff's theorem that $$ \lim_{n \rightarrow \infty} \frac{f_n}{...
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      How to find the cosets parameterizing quotient spaces such as elliptic three-manifolds?

      I would like to know, given a groups $H\subseteq G$, if there a general method to finding the representatives $g$ that would parameterize the cosets $gH$. I cannot find a recipe for infinite $G$ and ...
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      60 views

      Can Gaussian measure be characterized by unitary representations?

      It is well known that Fourier transform switches positive-definite functions with positive measures on a (locally compact topological) group. Further, the positive definite functions can be ...
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      77 views

      Singular integral operators and PDEs

      What is the relation between the notion of singular integral operators and partial differential equations? I know, for example, that there is a relation between the Cauchy transform (Riesz transforms ...
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      30 views

      Fourier Restriction: extension operator identity

      Consider the extension operator: $$ Eg(x)=\int_S g(\xi)e^{2\pi i x\cdot \xi}d\sigma(\xi). $$ For simplicity we consider the 2D-case, where $S$ is the paraboloid $\xi\mapsto \xi^2$, $\xi\in [-1,1]$. (...

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