# 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.

1,445
questions

9
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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 ...

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-...

2
votes

1
answer

60
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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 ...

2
votes

1
answer

228
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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 ...

3
votes

1
answer

182
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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. ...

0
votes

0
answers

18
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### 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\...

1
vote

0
answers

31
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### $H^1$ 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 $(p,p)$ ...

2
votes

0
answers

173
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### 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 ...

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-...

-2
votes

1
answer

108
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### 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 ...

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 ...

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 $A$ ...

2
votes

1
answer

159
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### 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 ...

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 $X$ ...

6
votes

0
answers

124
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### 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 ...

4
votes

1
answer

186
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### 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 ...

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}(\...

10
votes

1
answer

629
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### 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 ...

0
votes

0
answers

36
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### 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:
$Z_0=X$, $...

6
votes

1
answer

228
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### 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^\...

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}...

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 ...

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 ...

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_{\...

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}...

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 ...

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 \...

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
$$
\|...

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 ...

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)} \...

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. ...

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 ...

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 ...

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 ...

6
votes

0
answers

137
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### 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^{...

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 ...

-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\,...

1
vote

0
answers

57
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### 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

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. ...

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 ...

2
votes

1
answer

303
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### 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

118
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### 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 ...

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 ...

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 ...

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}...

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 $[0,\lambda]$-...

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 ...

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 $p=\pi/2$...

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*...

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 ...