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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Are all enveloping algebras $\mathcal{U}(\mathfrak{g})$ locally compact quantum groups?
Let us consider the enveloping algebra $\mathcal{U}(\mathfrak{g})$ of some Lie algebra $\mathfrak{g}$.
Under what assumptions about $\mathfrak{g}$, does the enveloping algebra generate a locally ...
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Carleson's theorem: proof of a lemma
I am reading the paper of Michael Lacey called "Carleson's theorem: proof, complements, variations" 1, on Carleson's theorem in Fourier analysis. At the bottom of page 20 at the beginning of ...
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What is the fastest algorithm for classical period finding?
Let $N$ be a positive integer, and choose an integer $a$ such that $\gcd(a,N)=1$. Then $a^r \equiv 1 \,\text{mod}\, N$ for some $r$. What is the current fastest classical algorithm for finding the ...
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Proving a Fourier transform inequality for functions with mixed variable bounded support
I'm working on a problem involving the Fourier transform and have encountered an inequality that I am unsure how to prove. I would greatly appreciate any help or guidance you can provide.
Let $\gamma\...
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measure of Haar
Let $(G,K)$ be a Gelfand pair.
Why, for a function $f$ $K$-binvariant with respect to a compact subgroup $K$ of a group $G$, do we have the following equality:
$$ f(xy) = \int_K f(xky) \, dk$$
A ...
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Representation and Laplacian on the Heisenberg group
Let $\pi_{\lambda}$ be the the Schrödinger representations of the Heisenberg group $H=\mathbb C\times \mathbb R$. For $\varphi\in L^2(\mathbb R)$, we have
$$\pi_{\lambda} (z,t)\varphi(\xi)=e^{i\...
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Recognizing when a $2\pi$-periodic function is a shifted sine
Consider a real analytic function $f(x)$ with a period of $2\pi.$ Let $t\in \operatorname{Range}(f(x))$, $$I(t)=\int_{0}^{2\pi}\log |t-f(x)|dx.$$ Prove that $I(t)$ is equal to a constant if and only ...
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Is there an integer sequence $(k_n)$ where each term is non-zero such that $\lim z^{k_n}=1$ for every point z on the unit circle?
Is there an integer sequence $(k_n)$ where each term is non-zero such that $\lim z^{k_n}=1$ for every point $z$ on the unit circle of the complex plane? I don't think it exists，but I don't know how to ...
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Morphisms on L^2(G) induced by morphisms of LCA groups
I am looking for a good reference to understand the space $L^2(G)$ for a locally compact abelian (LCA) group $G$.
In particular, I would like to understand when $L^2(-)$ is functorial, so that if $\...
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Plancherel's Theorem in wider space
By Planchere's Theorem, we have for $f$, $g$ in $L^2(\mathbb{R}^n)$ we have
$$\int_{\mathbb{R}^n}fg=\int_{\mathbb{R}^n}\hat{f}\hat{g}$$
Also, by distribution theorey, we can define the Fourier ...
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Peter–Weyl decomposition of a group representation rather than group algebra
Consider a finite or compact group $G$. The Peter–Weyl decomposition is usually formulated for the group algebra $\mathbb{C}[G]\simeq\bigoplus_i \operatorname{End}(V_i)$, where $V_i$ are the spaces of ...
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What are the finite-dimensional irreducible unitary representations of $E(3)$?
Let $E(3)$ be the Euclidean group of $\mathbb{R}^3$ defined, e.g., by
$$E(3)=SO(3)\ltimes T(3)$$
where $T(3)$ is the translation group.
I am looking for a reference classifying all the finite-...
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Locally compact groupoid with a Haar system such that the range map restricted to isotropy groupoid is open
Can somebody provide an example of a locally compact groupoid $G$ with a Haar system such that the range map restricted to isotropy groupoid of $G$ is open?
I could not find any specific example for ...
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Asymptotics of an oscillatory integral
For $a > 0$ and $n \in \mathbb Z_+$, consider the oscillatory integral
$$\int_{0}^1 f(x) f(ax) \dots f(a^n x) \, dx,$$
where $f$ is an integrable function on $[0, 1]$, which we extend by ...
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A formula resembling the integral mean value on Kähler manifolds
I am reading comparison geometry of holomorphic bisectional curvature for Kähler manifolds and limit spaces by John Lott. I have a problem when reading the proof of the following theorem:
Theorem. ...
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