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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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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 ...
Conifold's user avatar
  • 1,711
2 votes
0 answers
84 views

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-...
PontyMython's user avatar
2 votes
1 answer
60 views

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 ...
K N Sridharan's user avatar
2 votes
1 answer
228 views

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 ...
Nate River's user avatar
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3 votes
1 answer
182 views

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. ...
HeroZhang001's user avatar
0 votes
0 answers
18 views

Is there a classification of 2D projective convolution kernels?

Is there any classification of all distributions on $\mathbb{R}^2$ such that they are equal to the convolution with themselves? i.e. given a distribution $\gamma$ under which conditions $$ \gamma\star\...
Nicolas Medina Sanchez's user avatar
1 vote
0 answers
31 views

$H^1$ and $BMO$ interpoaltion on spaces of homogeneous type

In this Coifman and Weiss I've found that in spaces of homoegeneous type holds a $(H^1,L^1)-(L^p,L^p)$ interpolation theorem. (it is even stronger because it only needs weak $(H^1,1)$ and weak $(p,p)$ ...
Luca.b's user avatar
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2 votes
0 answers
173 views

How can the maximal ideal space of the Fourier Stieltjes algebra be non-separable?

I have been asking a fair few (probably elementary) questions about abstract harmonic analysis lately. By means of explanation, I am just feeling around the subject at the moment and trying to build ...
Daron's user avatar
  • 1,915
1 vote
0 answers
26 views

Completeness of the atomic Hardy space $H^{p,q}(X)$ on spaces of homogeneous type

I have a problem with the following fact: the article by Coifman and Weiss, "Extensions of Hardy spaces and their use in analysis" https://projecteuclid.org/journals/bulletin-of-the-american-...
Giovanni Marano's user avatar
-2 votes
1 answer
108 views

Mismatch between equivalent definitions of the Bohr compactification of the reals

I feel I'm overlooking something very silly. The Bohr compactification of $\mathbb R$ has two equivalent definitions. The set of (possibly discontinuous) homomorphisms $\mathbb R \to \mathbb T$ under ...
Daron's user avatar
  • 1,915
4 votes
3 answers
357 views

Fourier transform in $L^1$?

Let $f \in L^1 \cap L^2$. Are there any natural conditions on $f$ that ensure that the Fourier transform $\hat f$ is in $L^1?$ I don't want to have anything as restrictive as Schwartz. I am rather ...
António Borges Santos's user avatar
1 vote
0 answers
92 views

Let $A:=\{f\in C^1(\mathbb{R}): \hat{f}, \hat{f'} \in L^1(\mathbb{R})\}$. Schwartz space is dense in $A$ wrt $\|f\|:= \|\hat{f}\|_1+\|\hat{f'}\|_1$?

Let $A:=\{f\in C^1(\mathbb{R}): \hat{f}, \hat{f'} \in L^1(\mathbb{R})\}$, where $\hat{f}$ is the Fourier transform of $f$. Then is it true that Schwartz space $\mathcal{S}(\mathbb{R})$ is dense in $A$ ...
mathlover's user avatar
2 votes
1 answer
159 views

Does $i\partial_tu = \Delta^2 u$ exhibit more or less dispersion than $i\partial_t u= \Delta u$?

Consider the initial-value problems in $d=1$ $$\begin{cases} i\partial_tu = \Delta^2 u \\ u(x,0)=u_0 \end{cases}$$ and $$\begin{cases} i\partial_t u= \Delta u \\ u(x,0)=u_0, \end{cases}$$ Solutions to ...
Diffusion's user avatar
  • 763
1 vote
2 answers
217 views

Reverse Markov inequality

Let $C > c > 0$ and $K > 1$ be constants. Does there exist, for all small enough $\varepsilon > 0$ depending on $c, C, K$, some bound of the following form? For all random variables $X$ ...
Nate River's user avatar
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6 votes
0 answers
124 views

Are there isospectrally equivalent exotic spheres?

Let $X$ and $Y$ be two different exotic spheres. Are there metrics $g$ and $h$ on $X$ and $Y$, respectively, such that the laplacians of $(X,g)$ and $(Y,h)$ have the same spectrum? I would be happy ...
discretephenom's user avatar
4 votes
1 answer
186 views

estimate a singular integral using a dyadic decomposition

Let $0<\alpha_{j}<1$, $j=1,\dots,d+1$. I am trying to estimate the following singular integral: $$I(y_{1},\dots,y_{d},z):=\int_{\substack{ x\in[0,1]^{d}\\ 1/2<|x|<1}} \frac{d x_{1} \dots d ...
Medo's user avatar
  • 784
1 vote
1 answer
103 views

How to show such result for generalized $ O(|x|^{-1/2}) $ function?

Assuming that $ \chi\in C_c^{\infty}([-2,2]) $ is a cutoff function such that $\text{supp }\chi\subset[-2,2]$, $\chi\equiv 1 $ in $ [-1,1] $, and $ 0\leq\chi\leq 1 $, suppose that $ f\in C^{\infty}(\...
Luis Yanka Annalisc's user avatar
10 votes
1 answer
629 views

On Pareto functions

The Pareto principle says that the top 20% of wealthy people people hold over 80% of the wealth. Suppose we had a non-negative function on $\mathbb R^n$ that satisfied this principle on every open ...
Nate River's user avatar
  • 5,409
0 votes
0 answers
36 views

Sufficient condition for interpolation

If we have a couple of two compatible banach spaces (in this sense) $(X,Y)$ and a sequence of Banach spaces $\{Z\}_{\theta\in[0,1]}$ which are intermediate between $X$ and $Y$ satisfying: $Z_0=X$, $...
mejopa's user avatar
  • 101
6 votes
1 answer
228 views

Poisson kernel for the orthogonal groups

For the complex ball $|z|^2\le 1$ in $\mathbb{C}^n$, there is a Poisson kernel proportional to $|x-z|^{-2n}$. This is generalized to the unitary group $U(N)$ so that in the complex matrix ball $Z^\...
thedude's user avatar
  • 1,509
0 votes
0 answers
83 views

Maximal function on mixed $L^{p}$

Consider $ f_{j,k}$ to be a function in $L^{p}(l^{q}(l^{2}))$, that is $$ \Vert f_{j,k} \Vert^{p}_{L^{p}(l^{q}(l^{2}))} = \int_{\mathbb{R}^{n}} \left( \sum_{k} \big[ \sum_{j} \vert f_{j,k}(x) \vert^{2}...
User091099's user avatar
0 votes
1 answer
113 views

Clarification on the Interpretation of Fourier Coefficients in the Context of Fourier Projections

I am currently studying a paper (Section 3.4.3 of Lanthaler, Mishra, and Karniadakis - Error estimates for DeepONets: a deep learning framework in infinite dimensions) where the authors define an ...
Mohammad A's user avatar
2 votes
1 answer
77 views

Explicit computation for the coefficients of the intertwining operator

In the following note by Casselman https://personal.math.ubc.ca/~cass/research/pdf/p-adic-book.pdf page 74, proposition 9.3.2, for the group $G=SL_{2}$ he computed explicitly the coefficients of the ...
R. Chen's user avatar
  • 121
4 votes
1 answer
157 views

Uniform decay of $J'_{\nu}(x)$ for $x\gg1$

I need a uniform decay estimate for the derivative $J'_{\nu}(x)$ of the Bessel functions. By `uniform' I mean an estimate independent of $\nu$, at least for a range of orders like $\nu\ge0$. For $J_{\...
Piero D'Ancona's user avatar
2 votes
0 answers
27 views

Dual of homogeneous Triebel-Lizorkin

Let $ p, q \in (1,\infty)$ and consider the homogeneous Triebel- Lizorkin space $\dot{F}^{s}_{p,q}$ to be the space of all tempered distributions (modulo polynomials) with $$ [f]^{p}_{\dot{F}^{s}_{p,q}...
User091099's user avatar
1 vote
1 answer
71 views

Why are symmetric convex bodies with a smooth boundary and non-vanishing Gaussian curvature of particular interest in harmonic analysis?

I don't work in harmonic analysis or convex analysis, but in some literature of harmonic analysis, I often see the assumption that "let $K$ be a symmetric convex body with a smooth boundary and ...
taylor's user avatar
  • 445
1 vote
0 answers
53 views

Reference for Density question

Let $ B $ be a reflexive, separable Banach space and $ p \in (1,\infty)$. Then denote by $L^{p}(B)$ the space of all functions $$ f : \mathbb{R}^{n} \to B $$ with $$ \int_{\mathbb{R}^{n}} \vert f \...
User091099's user avatar
1 vote
1 answer
107 views

approximating differentiable functions with double trigonometric polynomials

Let $Q = [0,1]^2$. For sake of notation, let $$ f^{(i,j)}(x,\xi) = \frac{\partial^{i+j}}{\partial x^i \partial \xi^j}f(x,\xi). $$ Fix some non-negative integer $k$. Moreover let $f\in C^k(Q)$ if $$ \|...
Doofenshmert's user avatar
10 votes
3 answers
1k views

What is the intuition behind applying the Mellin Transform to prime distribution?

I am an undergraduate student writing an expository thesis on the complex-analytic proof of the Prime Number Theorem. I understand that applying the Mellin Transform to the partial sum of the van ...
onionbread's user avatar
0 votes
2 answers
223 views

Does this distribution exist?

Assume there is a distribution in two variables $\mathcal{W}\in\mathcal{S}'(\mathbb{R}^2)$ with Fourier transform $\hat{\mathcal{W}}(\alpha,\beta)\equiv \int_{-\infty}^\infty e^{i(\alpha x+\beta y)} \...
Nicolas Medina Sanchez's user avatar
1 vote
1 answer
139 views

Radial Fourier transform vs Hankel transform

I was wondering about the relationship between the Hankel transform and the Fourier transform for radial functions. Let $f : \mathbb{R}^d \to \mathbb{R}$ be a (sufficiently regular) radial function. ...
Chris Jones's user avatar
0 votes
0 answers
33 views

Reference request: injectivity of CWT, density of dilations and translations in $L^p$

Recently, I encountered the notion of Continuous Wavelet Transform (CWT), and I find it very intriguing (for a reference, see the wiki). I believe it offers a different and more general perspective on ...
Zhang Yuhan's user avatar
0 votes
0 answers
63 views

Multiplication with dilations of nonzero measurable function is injective

Denote $f_s(x):=f(sx)$ as the dilation of a function $f$. I want to know whether the following statement is true: Suppose $f$ and $g$ are measurable functions on $\mathbb{R}$, and $f$ is not almost ...
Zhang Yuhan's user avatar
7 votes
1 answer
305 views

High dimensional Lusin conjecture

Lusin, in 1913, while considering the properties of Hilbert's transform, conjectured that every function in $L^2[-\pi, \pi]$ has an a.e. convergent Fourier series. Kolmogorov, in 1923, gave an example ...
Tomas's user avatar
  • 869
6 votes
0 answers
137 views

Estimating the Hausdorff dimension of the discontinuity set of a function

Suppose $\sum_{\xi \in \mathbb{Z}^d}{a_{\xi}}e^{i\langle x, \xi\rangle }$ converges spherically pointwise to $0$ for all $x \in \mathbb{T}^d$, i.e. $\lim_{R \to \infty} \sum_{|\xi| < R}{a_{\xi}}e^{...
fwd's user avatar
  • 161
3 votes
0 answers
126 views

Explicit basis of symmetric harmonic polynomials

An orthonormal basis for the space of harmonic polynomials in $n$ variables is provided by the spherical harmonics on the $n-1$ sphere, see e.g. wiki. From there, constructing an orthonormal basis for ...
Cacuete's user avatar
  • 31
-1 votes
1 answer
281 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\,...
Z. Alfata's user avatar
  • 640
1 vote
0 answers
57 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 $L^2(M)$ ...
Ali's user avatar
  • 4,103
2 votes
1 answer
105 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. ...
Ali's user avatar
  • 4,103
1 vote
0 answers
41 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 ...
Timothy Chu's user avatar
2 votes
1 answer
303 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 \...
Zhang Yuhan's user avatar
1 vote
1 answer
118 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 ...
Xin Qian's user avatar
  • 145
4 votes
1 answer
260 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 ...
António Borges Santos's user avatar
0 votes
1 answer
123 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 ...
António Borges Santos's user avatar
2 votes
0 answers
49 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}...
Giulio Binosi's user avatar
2 votes
0 answers
76 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 $[0,\lambda]$-...
an_ordinary_mathematician's user avatar
5 votes
1 answer
240 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 ...
EG2023's user avatar
  • 63
3 votes
0 answers
172 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 $p=\pi/2$...
AlphaHarmonic's user avatar
2 votes
1 answer
121 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*...
an_ordinary_mathematician's user avatar
0 votes
1 answer
113 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 ...
Ma Joad's user avatar
  • 1,683

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