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Results tagged with harmonic-analysis
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user 10400
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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Background for Hejhal's "The Selberg Trace Formula for $PSL(2, \mathbb{R})$
Deitmar and Echterhoff "Principles in Harmonic Analysis" Chapter 9 and 11 for the cocompact case (Hejhal I). It requires some familiarity with representation theory, but you seem to be more interested …
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7
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3
answers
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A Question concerning the Fourier Transform of $\mathbb{R}$
Consider the classical Schwartz space $\mathcal{S}(\mathbb{R})$ together with the Fourier transform $\mathcal{F} : \mathcal{S}(\mathbb{R}) \rightarrow \mathcal{S}( \mathbb{R})$.
Consider the subspace …
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1
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Does random matrix theory make any prediction for the eigenvalue distributions of compact Ri...
Under RH, Montgomery has proven equidistribution results for the zeros of the Riemann Zeta function, which suggest a close connection of the distribution to certain results in Random matrix theory. An …
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3
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1
answer
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Inducing from cocompact subgroups
Consider a locally compact group $G$ and a cocompact subgroup $H$, is it known that the induction of an irreducible representation $\pi$ of $H$ to $G$ decomposes discretely into a direct sum of irredu …
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3
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Accepted
When does a LCA group not contain a (closed) infinite cyclic subgroup?
In general, you have for a compactly generated group $G = \mathbb{R}^n \times \mathbb{Z}^n\times K$, with $K$ compact. And there is no way to embed $\mathbb{Z}$ discretely in something compact, see D …
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3
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1
answer
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Ergodic decomposition of quasi-invariant measure
I have a reference request concerning Proposition 1.6 in the following article Link
The setting: Let $G$ be a locally compact, second countable group. Let $S = (S, \mu)$ be a Polish space. Assume we h …
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Topology on the Unitary Dual
Convergence in the Fell topology is equivalent to convergence of matrix coefficients. In the finite-dimensional case, this is equivalent as $\rho_n(g) v \rightarrow \rho(g)v$.
Quote from Vogan (http: …
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3
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2
answers
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Measures and structure on conjugacy classes
Given a locally compact group $G$, does there exist a measure $\nu$ on the conjugacy classes $conj(G)$ such that for $f \in C_c(G)$
$$ \int_G f(g) d \mu_G(g) = \int_{conj(H)} \int_{G / G_\gamma} f(g\g …
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2
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Where do the real analytic Eisenstein series live?
I can give an additional point of view, which is coming from the theory of parabolic induction. Parabolic induction plays a prominent role in representation theory and gives you a better intuition for …
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Decomposition of $L^2(\Gamma \backslash G)$
The continuous part(=Eisenstein series) is understood by Moeglin-Waldspurgers book on Eisenstein series.
The cuspidal part(=discrete part) is not well understood. Many things are still open. I am re …
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Accepted
Spherical functions for sl(2,Q_p)
Be careful that among the irreducible unitary reps also the trivial representation has this property. That's why Paul Garrett says "embeds" into a prinicpal series, so you get not only unramified unit …
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How does one show the existence of discrete and complementary series for SL(2,R)?
I found Traces of Hecke operators by Knightly and Li very readable. They treat Gl(2,R) by a similar method. Knapp or Wallach is
also nice to read and more general. They have chapters for Sl(2,R) and …
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The measure on the harmonic spectrum from Selberg trace formula
The physics paper regularizes the volume and I don't expect a straight forward translation between the Selberg trace formula setting for finite volume Riemann surface and the regularized upper halfpla …
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Support of functions on compact groups, and their Fourier transforms
Certainly you can recover $f$ for a circle, because the group is abelian. In general, $f$ can be written as a linear combination of matrix coefficients, and for the Fourier transform of a general comp …
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Orthonormal basis for $L^2(G/H)$.
This was merely too long for a comment.
There is a certain preferable choice in a special situation, where one can choose an orthornormal basis of a rather specific nature.
Let $H$ be cocompact in $ …
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