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.
1,413
questions
3
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Average size of the Fourier--Stieltjes transform of the fractal measures
For $0<\theta<1/2$ define $\mu_\theta$ to be the uniform (self-similar) measure on the Cantor set obtained from the dissection pattern $(1-2\theta,\theta)$. For example, when $\theta=1/3$ the $\...
1
vote
1
answer
118
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estimate involving Gaussian data
Let $x\in \mathbb{R}^{d}$, $d\geq 1$ and $p>\frac{2}{d}$. For any $R>0$ and any $0<s<\frac{dp}{2}-1$
\begin{align}
&\int_{\mathbb{R}}\int_{|x|>R}\left(\frac{1}{1+t^2} \right)^{\frac{...
2
votes
0
answers
58
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A lemma in the application of Lions's concentration compactness pricnciple in Hardy-Littlewood-Sobolev inequality
I'm encountering some problems when reading Lions' paper "the concentration-compactness principle in the calculus of variations. The limit case, Part 2".
The Hardy-Littlewood-Sobolev (HLS) ...
5
votes
1
answer
235
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Is the local maximal function bounded from $W^{1, 1}$ to $L^1$?
Let $f \in W^{1, 1} (\mathbb R^d)$. For every $\varepsilon > 0$, we consider the local maximal function $M_\varepsilon f: \mathbb R^d \to \mathbb R$, defined by
$$M f_{\varepsilon} (x) = \sup_{r \...
0
votes
0
answers
74
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Characterising wavelet frames using Fourier transform
As usual, for $f\in L^2(\mathbb R)$
$$
D_j(f)(x) = 2^{j/2} f(2^jx), T_k(f)(x) = f(x-k)
$$
and let $\{D_j T_k \varphi\}$ be a system of wavelets on $\mathbb R.$
A simple result is that, $T_k \varphi$ ...
4
votes
1
answer
450
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Showing that decay results of Fourier coefficients are sharp
I originally post the question here but the it seems too advanced for undergraduate level. At least, an example is very hard to find, so I am putting it here.
It is well-known that if $f$ is ...
2
votes
1
answer
221
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Examples of non-discrete, cocompact subgroups
I am looking for non-trivial examples of the following:
$G$ is a locally compact group
$H\subset G$ a closed subgroup
Both are unimodular and non-discrete
The quotient space $G/H$ is compact, but $G$ ...
4
votes
1
answer
738
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How to get $\int_{\mathbb R^d} |\partial_i\partial_j(1-\Delta)^{-\frac{\delta}{2}}p_t(\cdot-y)(x)| \, \mathrm d x \lesssim t^{\frac{\delta}{2}-1}$?
We consider the Gaussian heat kernel $p_t$ on $\mathbb R^d$, i.e.,
$$
p_t (x) := (4\pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{4t}},
\quad t>0, x \in \mathbb R^d,
$$
and define the operator $P_t$ by
$$
...
3
votes
1
answer
275
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Heat kernel of left-invariant metric on 3-sphere
This post concerns the heat equation on $S^3 (\simeq \mathrm{SU}(2))$ endowed with an arbitrary left-invariant metric. We think of $S^3$ as the space of unit quaternions, and its Lie algebra $\...
5
votes
1
answer
431
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If $f$ is bounded, decays fast enough at infinity and $\int f=0$, does this imply that $f$ is in the Hardy space $\mathcal H^1(\mathbb R^n)$?
Let $\mathcal H^1(\mathbb R^n)$ be the real Hardy space (as in Stein's "Harmonic Analysis", Chapter 3). It is well known that $\mathcal H^1(\mathbb R^n)\subset L^1(\mathbb R^n)$ and its ...
1
vote
0
answers
64
views
Eigenvalue problem of The Dirichlet problem
Consider the nonlocal problem
\begin{aligned}
&u_t(x,t)=\int_{\mathbb{R^n}}J(x-y)u(y,t)dy - u(x,t),&x\in\Omega,t>0,\\
&u(x,t)=0&x\notin\Omega,t>0,\\
&u(x,0)=u_0(x).&x\in\...
5
votes
3
answers
300
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The integrability of $\widehat{e^{-|x|^a}}$, $a>0$
Let $f_a(x)=e^{-|x|^a}$, $x\in \mathbb{R}^n$. Then $f \in L^p(\mathbb{R}^n)$ for every $1\leq p\leq \infty$. It is also smooth away from the origin and decays faster than any polynomial as $|x|\...
0
votes
1
answer
533
views
Does this dyadic sum converge?
Let $a\in (0,1)$ and define
$$J(j):=\int_{0}^{\infty} e^{- 2^{j} s} \frac{s^{a}}{1+s^{2a}} ds,\quad j\in \mathbb{Z}.$$
Note that rescaling $2^{j} s\mapsto s$ shows that
$$J(j)\leq 2^{-j(1+a)}\int_{0}^...
2
votes
1
answer
152
views
Continuity of an upper semi-continuous function over periodic points
Let $f: X \to \mathbb{R}$ be an upper semi-continuous function on $X$, which is a compact subspace of a vector space. Let sequence $x_n, n \in \mathbb{N}$, with positive elements - periodic: there ...
1
vote
0
answers
62
views
A kind of weak convergence for Sobolev spaces with zero on boundary
Let $\Omega \subseteq \mathbb{R}^d$ be bounded and open with $\partial\Omega$ Lipschitz, $1<p<\infty$. Is it true that if $(\varphi_n)_{n\geq 1}\subset C^{\infty}_c(\Omega)$ with $\varphi_n\to \...
2
votes
0
answers
409
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Generalized conjugacy classes in (topological) groups
Let $G$ be a topological group. We define an equivalence relation on $G$ as follows:
For $a,b\in G$ we set $a\sim b$ if the following two maps are topologically conjugate:
$$x\mapsto ax,\qquad x\...
2
votes
0
answers
161
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Approximating $L^p$ functions by eigenfunctions of Laplacian
I'm reading a paper https://www.sciencedirect.com/science/article/pii/S0022039608004932.
In this paper, the authors assume that $\mathcal{O}$ is a bounded domain of $\mathbb{R}^N$ with $C^m$ boundary ...
1
vote
0
answers
147
views
The space of ergodic elements of a topological or Lie group
Let $G$ be a compact topological group with normalized Haar measure $\mu$. An element $g\in G$ is an ergodic element if the mapping $L_g:G \to G $ with $x\mapsto gx$ is an ergodic map. The ...
1
vote
2
answers
177
views
Solution of $\Delta f -\frac{1}{2}hf = 0$ behaves asymptotically as $f(x) = 1 - C/|x|$
Let $f: \mathbb{R}^{3} \to \mathbb{R}$ be the solution of the following PDE:
$$\Delta f -\frac{1}{2}h f = 0$$
where $h \in C_{c}^{\infty}(\mathbb{R}^{3})$ (compactly supported an smooth) and $f$ ...
4
votes
1
answer
170
views
When is $W^{1,p}(\Omega)$ a Banach algebra?
Let $\Omega \subseteq \mathbb{R}^d$ be bounded and open with $\partial\Omega$ Lipschitz, $1<p<\infty$.
My question: knowing that $f,g\in W^{1,p}(\Omega)$ for what $p$ can we conclude that $f\...
3
votes
0
answers
115
views
$L^\infty-L^1$ norm of an oscillatory integral operator
Let $\lambda>1$, $x,y\in [0,1]$, and $K_\lambda(x,y)=(1+\lambda|x-y|)^{-2}e^{i\lambda|x-y|}$. Consider the operator
$$T_\lambda f(x)=\int_0^1 K_\lambda(x,y)f(y)dy.$$
We want to precisely estimate ...
3
votes
0
answers
48
views
Matrix argument K Bessel functions at half integral orguments
As a working definition I will define:
$$
K_\nu^{(n)} (z) = \frac{1}{2^n}\int_{\mathcal{P}} e^{- \operatorname{tr}( y + y^{-1}) z/2} \det y^\nu d \mu(y)
$$
where $\mathcal{P}$ represents the space of ...
15
votes
4
answers
1k
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Steinhaus theorem and Hausdorff dimension
Assume for simplicity that sets $A_i\subset\mathbb{R}$ are compact. If $A_1$ and $A_2$ have positive length, then $A_1+A_2$ contains an interval. That is a variant of the classical Steinhaus theorem ...
4
votes
1
answer
189
views
Fourier coefficients of Selberg polynomials
In Montgomery's "Ten Lectures on the Interface Between Number Theory and Harmonic Analysis" a bound for the Fourier coefficients of the Selberg polynomial $S^+_K$ is obtained by using what ...
-1
votes
1
answer
121
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Riesz energy for open sets in dimension $1$
This is a continuation of the question Calculation of Riesz energy for balls . As there are three questions,;I am posting a new question here. Riesz energy for a ball $B(x_0,r)$ is given by
$$I_s(B(...
2
votes
0
answers
276
views
Recent progress restriction conjecture - Problem 2.7 (Terence Tao lecture notes)
I've been tackling the following problem for some time,
Problem 2.7. (a) Let $S:=\left\{(x, y) \in \mathbf{R}_{+} \times \mathbf{R}_{+}: x^2+y^2=1\right\}$ be a quarter-circle. Let $R \geq 1$, and ...
8
votes
2
answers
604
views
Uniqueness of the uniform distribution on hypersphere
I'm looking for a uniqueness-type result for the following problem, which is related to the uniform distribution in the hypersphere $\mathbb{S}^{p-1}$. Suppose $f$ is a sufficiently smooth function on ...
1
vote
0
answers
90
views
Showing Vaaler polynomial is a good approximation to saw tooth function
Vaaler's polynomial is defined
$$
V_K(x) = \frac{1}{K+1}\sum_{k=1}^K\left(\frac{k}{K+1} - \frac12\right) \Delta_{K+1}\left(x - \frac{k}{K+1}\right) +
\frac{1}{2 \pi (K+1)}\sin 2 \pi (K+1) x - \frac{1}{...
2
votes
1
answer
130
views
Orthonormal bases in RKHSs via interpolating sequences
Definitions and setting
Let $\mathcal{H}$ be a separable, infinite-dimensional, reproducing kernel Hilbert space on a nonempty set $X$. As usual, denote the reproducing kernel on $\mathcal{H}$ by $K$ ...
6
votes
1
answer
191
views
Is the Hardy Littlewood “minimal function” comparable to the original function in $L^1$ norm?
Given $f \in L^1 (\mathbb R^d)$, and $\varepsilon > 0$, define the minimal function $m_\varepsilon f$ by
$$m_\varepsilon f(x) := \inf_B \frac1{|B|} \int_B |f| ,$$
where the infimum is taken over ...
3
votes
1
answer
189
views
Eigenforms of the Laplacian on Lie groups
I am a bit rusty in my differential geometry and I would like to confirm that my reasoning below holds, and I have some related questions (and all references to related concepts are of interest to me)....
1
vote
0
answers
112
views
Idempotent conjecture and non-abelian solenoid
Is there a discrete non-abelian group whose dual in a reasonable sense is isomorphic to the solenoid constructed via a sequence of quaternions $S^3$ instead of a sequence of circles? The motivation ...
0
votes
0
answers
94
views
Idempotent conjecture and (weak) connectivity of (a reasonable) dual group
What is an example of a torsion free discrete abelian group $G$ whose dual space $\hat{G}$ is not a path connected space?
The Motivation: The motivation comes from the idempotent conjecture of ...
1
vote
0
answers
58
views
$L^p$ norm of Fourier transform of function composed with a diffeomorphism
Suppose $f$ is a compactly supported smooth function from $\mathbb{R}^n$ to $\mathbb{C}$ and $A$ is a diffeomorphism on $\mathbb{R}^n$, do we have any theorems relating the $L^p$ norm of $\hat{f}$ and ...
2
votes
1
answer
232
views
Maximal function on small cubes
I am reading "The Uncertainty Principle" by Fefferman (Bull. AMS, 1983) and have some issues following the arguments. In Lemma $C$ we have the following setting:
Let $Q^0\subseteq \mathbb{R}...
0
votes
0
answers
143
views
Why is this function in $L^1$?
I had a question about a claim made in the paper "Group Invariant Scattering" and why it is true. Consider the function $h_j(x) = 2^{nj}\psi(2^jx)$, where $\psi$ is a function such that $\...
2
votes
1
answer
194
views
Asymptotics for oscillatory integral
Consider the following integral for $f \in C_c^{\infty}(\mathbb R^n)$, $x_0$ fixed (possibly zero), and $n \ge 3$
$$F(\lambda) = \int_{\mathbb R^n} e^{i\lambda \vert x-x_0 \vert^2} \frac{f(x)}{\vert x ...
1
vote
0
answers
66
views
Estimating commutator of Fourier integral
Let $f(x)= \log(\vert x\vert)$ on $\mathbb R^2$ and define $s_n:H^2 \to L^2$ where $H^2$ is the second Sobolev space by
$$ s_n(g)(x) = \frac{nf(x)}{4\pi i} \int_{\mathbb R^2} e^{\frac{in\vert x-y\...
1
vote
0
answers
49
views
Are integration over restricted direct products only useful for specific functions?
So I've been reading Tate's thesis currently. In that we have defined integration of functions on $G$, which are basically formed from restricted direct products of locally compact groups $G_{\...
3
votes
2
answers
272
views
Calculation of Riesz energy for balls
I was reading stuffs about Riesz energy which is defined for an open subset $U\in\mathbb{R}^d$ by $I_s(U)=\int_U\int_U|x-y|^{-s}\ dx\ dy$ where $dx$ and $dy$ are Lebesgue measure in $U$. Now if I take ...
4
votes
1
answer
154
views
Uniqueness of critical points for Lipschitz perturbations of uniformly convex Hamiltonians
Consider a macroscopic free energy functional of the form
$$\mathcal{F}_\beta(\mu):= \frac{1}{\beta}\int_{\mathbb{R}^d}\log(\mu)\mu dx + \int_{\mathbb{R}^d}V(x)\mu(x)dx + \iint_{(\mathbb{R}^d)^2}g(x-y)...
2
votes
1
answer
85
views
Controlling the tensor product of functions in $H^1$ with lower derivatives
Given some function $\phi:D\to\mathbb{R}$, with $D\subseteq \mathbb{R}^d$. I was wondering whether we can find an estimate of the form
$$ \|\phi\otimes\phi\|_{\dot{H}^1(D\times D)} \lesssim \|\phi\|_{\...
8
votes
0
answers
168
views
What is known about when $vN(G)$ is a factor, for a locally compact group $G$?
When $G$ is a discrete group, it is an elementary result in the theory of von Neumann algebras that the group von Neumann algebra $vN(G)$ is a factor if and only if $G$ is an ICC group.
What is known ...
0
votes
0
answers
68
views
Oscillation of a polynomial
Recently I came across a statement in a paper that I am unable to verify. Namely, it roughly says that the oscillation of a polynomial on a cube can be controlled by the oscillation of the polynomial ...
3
votes
1
answer
330
views
Is there an operation in topology analogous to the operation of averaging over a compact subgroup in harmonic analysis?
Let me start with the following
Illustration: Let $G$ be a compact group, and let $\pi:G\to H$ be its (surjective) continuous homomorphism onto a (compact) group $H$. So we can think that $H$ is the ...
2
votes
0
answers
53
views
Wave equation time decay
I am trying to deduce the dispersive estimate for the free wave equation in $\mathbb{R}^{d+1}\equiv\{(x,t) : x\in\Bbb R^d \wedge t\in\Bbb R\}$ $$u_
{tt}-\Delta_xu=0$$ The fundamental solutions of this ...
3
votes
1
answer
76
views
Spectral disjointness of unitary representations of Type I groups and orthogonality
Background: If $\mathcal{H}$ is a Hilbert space and $U:\mathcal{H}\rightarrow\mathcal{H}$ is a unitary operator, then for each $f \in \mathcal{H}$ the sequence $(\langle U^nf,f\rangle)_{n = 1}^\infty$ ...
5
votes
1
answer
264
views
Maximal operator estimates for the Schrödinger equation
Let $a>0$ and consider the operator
$$Tf(t,x)= \int_{\mathbb{R}^{n}}e^{ i x\cdot \xi} e^{i t \lvert\xi\rvert^{a}} \widehat{f}(\xi) \, d\xi.$$
When $a=2$, the function $Tf$ solves the Cauchy problem ...
4
votes
0
answers
121
views
Distinguishing the Besov and Triebel-Lizorkin spaces
Theorem 2.3.9. in Triebel's Theory of Function Spaces states that the Besov space $B^{s_1, p_1}_{q_1} (\mathbb R^d)$ coincides with the Triebel-Lizorkin space $F^{s_2, p_2}_{q_2} (\mathbb R^d)$ if and ...
3
votes
1
answer
91
views
$L^\infty$ bound of $x^m \psi_n(x)$ where $\psi_n$ is a Hermite function and $m,n \in \mathbb{N}$ - extension from Cramer's inequality
For each $n \in \mathbb{N}$, the Hermite function $\psi_n : \mathbb{R} \to \mathbb{R}$ is a Schwartz function defined by
\begin{equation}
\psi_n(x):=(-1)^n(2^n n!\sqrt{\pi})^{-1/2} e^{x^2/2} \frac{d^n}...