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
0
votes
1
answer
126
views
Check an equation on the Heisenberg group $H_1$
The Heisenberg group $H_1$ is the set $\mathbb C\times \mathbb R$ endowed with the group law
$$ (z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right); \quad \forall z,w \in \mathbb C\,...
1
vote
0
answers
29
views
Density of zero modes
Let $(M,g)$ be a compact smooth Riemannian manifold with a smooth boundary. Let $\{(\lambda_k,\phi_k)\}_{k\in\mathbb N}$ be the spectral data on $(M,g)$, namely an orthonormal basis for $L^2(M)$ ...
2
votes
1
answer
79
views
On compactly supported functions with prescribed sparse coordinates
Let $\{\phi_n\}_{n=1}^{\infty}$ be an orthonormal basis for $L^2((0,1))$ consisting of Dirichlet eigenfunctions for the operator $-\partial^2_x + q(x)$ where $q \in C^{\infty}_c((0,1))$ is fixed. ...
1
vote
0
answers
38
views
If a weighted Laplacian's eigenfunction is zero in an open set, when is it identically zero?
Let $m, s \in ([0, 1]^d \rightarrow \mathbb{R}_{\geq 0}$). Define a weighted Laplacian $\Delta_{m, s}f$ evaluated at $x \in [0, 1]^d$ to be:
$m(x) \cdot \text{div} ( s(x) \nabla f(x))$.
What ...
1
vote
1
answer
2k
views
Monge–Ampère operator
I'm studying the article of Bedford–Taylor "Fine topology, Šilov boundary…" but I don't
understand the proof of the following proposition.
Let $u$, $v$ be plurisubharmonic functions defined ...
14
votes
2
answers
3k
views
Differentiability of Fourier series
Consider the function defined by the Fourier series
$$ f(x;\alpha) = \sum_{n=1}^\infty \frac{1}{n^\alpha} \exp(i n^2 x ) , $$
where $\alpha >1 $.
For what values of $\alpha $ is $f$ ...
2
votes
1
answer
280
views
Fourier series but different waveform
Given a nondegenerate smooth simple closed convex curve $f: [0,2\pi]\to \mathbb C \setminus \{0\}$ with winding number (around origin) $1$, and $f$ have zero mean. Let $f_n: [0,2\pi]\to \mathbb C \...
1
vote
1
answer
105
views
A question about the maximal function
Let $n>4$, $f\in C^{\infty}(\mathbb{R}^{n})$ and 0 denote the origin of $\mathbb{R}^{n}$. We define a weighted maximal function by $$Mf(x)=\sup_{0<r<1}r^{4-n}\int_{B_{r}(x)}|f|$$ which is ...
3
votes
1
answer
133
views
On self-duality of non-Archimedean local fields
The question to follow has already been asked by the OP at https://math.stackexchange.com/questions/3454735/on-self-duality-of-non-archimedean-local-fields. Due to a lack of feedback, the OP felt ...
-1
votes
0
answers
47
views
Literature search for results on Bochner-Riesz means for real functions
MathOverflow community!
I'm delving into a specific area of Fourier analysis and came across an intriguing lemma (referenced as Lemma 4 of Chen & Chen Paper) stating that for a compact set U in $...
-2
votes
0
answers
38
views
Is there a Bochner-Riesz Theorem for Real Functions with Each Summand Being a Product of Two Real Functions?
I am exploring the Bochner-Riesz theorem and its implications on real functions. Specifically, I'm curious whether there exists a version of the Bochner-Riesz theorem where each summand in the Bochner-...
4
votes
1
answer
259
views
Schroedinger operator in 2 dimensions with singular potential
Consider the Schroedinger operator
$$H = -\Delta + \frac{c}{\vert x \vert^2}$$
in two dimensions with $c >0$
This operator has a self-adjoint realization, since it is a positive symmetric operator ...
0
votes
1
answer
117
views
Singular integral bounded by Dirichlet form?
We define for some fixed $L$
$$\Omega:=\{(x_1,x_2) \in ([-L,L]^2 \times [-L,L]^2) \setminus \{x_1=x_2\}\},$$
in particular $x_1,x_2 \in \mathbb R^2.$
Let $f \in C_c^{\infty}(\Omega)$, then I am ...
2
votes
1
answer
464
views
Shift-invariant spaces
We can define a shift-invariant space as
$$V_{\varphi}(\mathbb{Z}):=\left\{\sum_{k\in\mathbb{Z}}c_k\varphi({\cdot}-k):(c_k)\in \ell_2\right\},$$
where convergence of the series is taken to be in $L^2(\...
2
votes
0
answers
44
views
Geometric explanation of Fueter-Sce-Qian Theorem and similar situations
In Clifford analysis there is a fundamental theorem due to Fueter and extended by Sce and Qian that says (in modern terminology) that the given a slice regular function $f:\mathbb{R}^{m+1}\to\mathbb{R}...
6
votes
4
answers
7k
views
Characterization of the non-negative definite functions $f(x,y)$
The common definition of the non-negative definite functions is as follows:
Definition 1: A continuous complex-valued function $f(x)$ is called non-negative definite, if for any real numbers $x_1,\...
2
votes
0
answers
62
views
Upcrossing lemma and subharmonic functions
I have been studying the upcrossing lemma for submartingales, which asserts that if $X_n$ is a non negative submartingale, and $
\lambda>0$ then if we denote by $U_n$ the number of $[0,\lambda]$-...
5
votes
0
answers
166
views
Example of an $H^1$ function on the bidisk that is not a product of two $H^2$ functions
Fix $n \in \mathbb{N}$ and consider the Hardy space $H^1 := H^1(\mathbb{D}^n)$, consisting of holomorphic functions $f$ on the unit polydisk $\mathbb{D}^n=\mathbb{D}\times\dots\times\mathbb{D}$ such ...
3
votes
0
answers
171
views
Maximum of an integral
Assume that $a>0$ and $r\in[0,1)$. How to prove that the function $$f(p)=\int_{-\pi}^\pi \left (1+r^2+2 r \cos x\right)^{a/2} |(2+a) \cos(x+p)-a r \cos(p)| \, dx$$ attains its maximum for $p=\pi/2$...
0
votes
1
answer
105
views
Controlling convolutions with maximal functions
For $f\in L^1(\mathbb R^n),$ let $Mf$ be the (Edited: changed the type of maximal function) Stein spherical maximal function. Let $\varphi\in C_c^\infty.$ Then, can we have a pointwise estimate of the ...
2
votes
1
answer
112
views
On a density property of signed singular measures
Suppose that $\mu$ is a signed finite Borel measure which is singular with respect to the Lebesgue measure in $[0,1]$. Is it true that there always exist a point $x\in [0,1]$ such that
\begin{equation*...
1
vote
1
answer
249
views
Reverse estimate on the Riesz potential $I_\alpha : L^{n/\alpha}\to \mathrm{BMO}$
$\newcommand\BMO{\mathrm{BMO}}$Consider the Riesz potential on $\mathbb{R}^n$ given by
$$
I_\alpha f(x) = c_{n,\alpha} \int_{\mathbb{R}^n} \frac{f(y)}{\lvert x-y\rvert^{n-\alpha}} dy.
$$
It is known ...
1
vote
0
answers
57
views
Harmonic analysis of vector bundles on symmetric spaces
This is a follow-up to my previous question.
Given a semisimple symmetric space $M\simeq G/H$, in particular, the real hyperbolic space $H_{p,q}\simeq SO(p,q)/SO(p,q-1)$, and a vector bundle $E$ over $...
0
votes
1
answer
111
views
Meaning of imaginary frequencies
in a formal calculation of frequencies from a time series I sometimes get purely imaginary frequencies, i.e. $\mathfrak{Re}(\phi)=0,\ \mathfrak{Im}(\phi)\ne 0$.
Question:
is there any mathematical or ...
0
votes
0
answers
59
views
Oscillatory integrals with critical points clustering on a submanifold of positive codimension
In Stein's manuscript "Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals", oscillatory integrals with discrete critical points are proven. I heard that you ...
1
vote
1
answer
109
views
An integrable estimate of the Hölder constant of the map $x \mapsto \int_{\mathbb R^d} f(y) \partial_1 \partial_1 g_t (x-y) \, \mathrm d y$
Let $(g_t)_{t>0}$ be the Gaussian heat kernel on $\mathbb R^d$, i.e.,
$$
g_t (x) := (4\pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{4t}},
\quad t>0, x \in \mathbb R^d.
$$
Let $f : \mathbb R^d \to \...
0
votes
0
answers
80
views
The asymptotic behaviour of the Fourier transform of a certain class of radially symmetric functions
Fix $\theta\in (-\pi/2,\pi/2)$ and let $a>0$. Suppose that $f:\mathbb{C}\rightarrow \mathbb{C}$ is analytic in $S:=\{z\in \mathbb{C}: |\arg{z}|<\pi/2\}$ and
$$|f(z)|\sim |z|^{-a},\qquad |z|\to \...
3
votes
1
answer
158
views
Equivalent Littlewood-Paley-type decompositions
The theory of Besov and Triebel-Lizorkin spaces usually proceeds by taking a dyadic decomposition of unity, i.e. some non-negative functions $\psi_0,\psi \in C_c^\infty(\mathbb{R})$ such that
\begin{...
12
votes
2
answers
1k
views
Entropy for Haar measure on $O(n)$
Let $G$ be a locally compact group. A measure $\mu$ is the right-Haar measure on $G$ if for every $g\in G$ and $E\subseteq G$ Borel set $\mu(Eg)=\mu(E)$. It is known that every locally compact group ...
0
votes
0
answers
32
views
On the I-method's energy increment calculation in a paper of Dodson
I am currently reading Dodson's Global Well-posedness for the Defocusing, Quintic Nonlinear Schrödinger Equation in One Dimension for Low Regularity Data article and I am trying to understand Theorem ...
0
votes
0
answers
115
views
Why for $\psi$ square-integrable function the zero mean condition is equivalent to $\hat{\psi}(0) = 0$?
I am studying the classical book "Ten Lectures on Wavelets" written by Ingrid Daubechies and I do not understand a specific point. I would appreciate it if someone could help me with ...
1
vote
0
answers
78
views
For any $\beta>0$, there is a constant $c>0$ such that $\left\|(1-\Delta)^{\frac{\beta}{2}} f\right\|_{\infty} \leq c\|f\|_{C_b^\beta}$
For any $n \in \mathbb{Z}^{+}$, let $C_b^n\left(\mathbb{R}^d\right)$ be the class of real functions $f$ on $\mathbb{R}^d$ with continuous derivatives $\left\{\nabla^i f\right\}_{0 \leq i \leq n}$ such ...
9
votes
1
answer
361
views
Relationship between Harish-Chandra Schwartz space and more generic Schwartz spaces
If $G$ is a connected semisimple Lie group with finite center, Harish-Chandra defined a Schwartz space of rapidly decreasing functions on $G$ as the space of $\mathrm{C}^\infty$ functions defined by ...
1
vote
0
answers
116
views
Multidimensional weighted Paley-Wiener spaces are Hilbert spaces?
How to rigorously demonstrate that multidimensional weighted Paley-Wiener spaces are Hilbert spaces?
I am utilizing the exponential type definition established by Elias Stein in the book 'Fourier ...
1
vote
1
answer
113
views
Can functions with "big" discontinuities be in $H^1$?
How can I prove that the function:
$$u:\Omega\to\mathbb{R},\ u(x)=\begin{cases} 0, x\in\omega \\[3mm] v(x), x\in\Omega\setminus\omega\end{cases}$$ is not in $H^1(\Omega)$, knowing that $v\geq 1$ is ...
0
votes
1
answer
77
views
An identity about Bessel potential operators
I'm reading this paper where I encounter below equality, i.e.,
$$
\begin{aligned} & \left|\int_{\mathbb{R}^d}\left\{\ell_s^2\left(a_s^{\gamma^2}-a_s^{\gamma^1}\right)_{i j} \partial_i \partial_j ...
5
votes
0
answers
125
views
Is $\mathbf{C}_p(X)$ self-dual?
Let $X$ be a set. Consider $\mathbf{Q}_p$ and $\mathbf{Z}_p$ as the $p$-adic numbers and $p$-adic integers, respectively. For any finite subset $F \subseteq X$, one can construct the topological ...
6
votes
1
answer
669
views
Fourier series of smooth functions in infinitely many variables
Let $J$ be a set (usually countable). Let $t_j$, $j\in J$, be variables in ${\mathbb R}/2\pi i{\mathbb Z}.$ Put $u_j=\exp(it_j),$ $j\in J.$ Introduce the following semi-norms on the space of Fourier ...
1
vote
1
answer
90
views
$l^2(L^p)$ Decoupling constant of congruent tubes
Demeter's book Fourier Restriction, Decoupling, and Applications give a principle that one cannot decouple in a direction where the manifold is flat. Which is the below proposition:
Proposition 9.5 ...
5
votes
0
answers
232
views
Is there a way to solve this integral on the sphere explicitly?
Let $k_{j}\in {\mathbb{Z}}^{+}$ and $\,a_{j}\in \,]0,1[$, be such that
$k_{j}\,a_{j}<1$, $j=1,\cdots,n$. Let $f:\mathbb{R}^{n}\rightarrow [0,\infty[$ be defined by the integral
$$f(y):=\int_{\...
5
votes
0
answers
105
views
A function whose derivatives belong to $BMO(\mathbb{R}^n)$
I am reading the paper "Bounded Mean Oscillation and Sobolev Spaces" by Robert
s. Strichartz, Indiana University Mathematics Journal , 1980, Vol. 29, pp.
539-558. In this paper he defines ...
4
votes
2
answers
589
views
When is the Fourier algebra $A(G)$ enough close to the Fourier-Stieltjes algebra $B(G)$?
I am reading P.Eymard's paper on the Fourier algebras of locally compact groups, and I have several questions about his constructions. I asked one of them in math.stackexchange, so far without success,...
1
vote
0
answers
56
views
Dispersive equations at low frequencies and time oscillations
It seems to me that nearly all the common linear dispersive equations have dispersion relations which vanish at the zero spatial frequency. For example:
The Schrodinger dispersion relation is $\omega(...
4
votes
0
answers
270
views
The convention of Fourier transform on symmetric spaces
When trying to understand the Plancherel formula of reductive symmetric space of Harish-Chandra class, I get confused on the convention of Fourier and related transforms.
$\newcommand{\H}{\mathcal{H}}
...
4
votes
0
answers
181
views
Bochner theorem for (non-abelian) discrete groups
I am interested in Pontryagin duality-like theories for discrete groups, more particularly, whether an analogue to Bochner's theorem for abelian groups exists in the discrete non-finite and non-...
3
votes
1
answer
154
views
Question regarding proof of Littlewood-Paley
I posted this question on Math.SE where I unfortunately received no answers even after a bounty. As such, I am putting it here, in hopes to receive a response.
For the proof of Theorem 6.1.6 in ...
2
votes
1
answer
153
views
Any references for generalised square functions?
In harmonic analysis, there is a big chunk of literature studying the square function $Sf=\|\{P_jf\}_{j=1}^\infty\|_{l^2}$, where $P_jf=(\psi_j\hat f)\check{}$ and $\{\psi_j\}$ is a partition of unity,...
15
votes
4
answers
1k
views
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 ...
11
votes
0
answers
2k
views
Eric T. Sawyer's proof of Fourier restriction conjecture
Some days ago Eric T. Sawyer uploaded a paper to arxiv claiming a proof of the Fourier restriction conjecture https://arxiv.org/pdf/2311.03145.pdf. If complete and correct this work will be a landmark ...
1
vote
0
answers
179
views
Is this a well known space? Perhaps homogeneous Sobolev-like space?
The homogeneous Sobolev space $\dot H^s(\mathbb{R}^n) $ is often defined as the closure of $\mathcal{S}(\mathbb{R}^n)$ under the norm
$$ || |\omega|^s \widehat{f} ||_{L^2(\mathbb{R}^d)} =\int_{\...