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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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Defining a Measure on Quotient Spaces
For general quotient, you will have to work with quasi-invariant measures. There is a multiplier, which is constant if the measure is invariant. For it to be constant, it is necessary and sufficient t …
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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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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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Traceable representation of reductive group over a p-Adic field.
There is a notion of a CRR group in Dixmier's book on $C^*$ algebras, which is a group such that for every unitary irreducible representation $\pi$, the integrated representation $\int \pi$, a represe …
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A question about the quotient measure on the ideles and the adeles
Let $F$ be a local field, then the additive Haar measure is easy to relate to the multiplicative Haar measure:
$$ \int_F f(x) d^+ x = \int_{F^\times} f(x) |x| d^+ x.$$
I know that the ideles have z …
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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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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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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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Plancherel formula for non-second-countable (non-unimodular) groups
Answer to the first question: Jacques Dixmier, Les C-algèbres et leurs représentations. Section 18.8.1
Comment on the second question: I actually believe a decomposition of von-Neumann algebra into …
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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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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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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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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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