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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What is standard continuity argument for well-posedness?

Motivation: I'm trying to understand the proof of Theorem 3.1 in Antonelli, Saut, and Sparber - Well-Posedness and averaging of NLS with time-periodic dispersion management. Though in the following I'...
 Analyst 's user avatar
4 votes
1 answer
553 views

The decay of Fourier coefficients and the continuity of functions

Let $ f $ be a function on $ \mathbb{T}=[0,1] $ ($ 1 $-periodic) with bounded variation. Prove that if $ \widehat{f}(k)=\int_0^1f(x)e^{-2\pi ikx}dx=o(1/|k|) $, then $ f\in C(\mathbb{T}) $. I do not ...
Luis Yanka Annalisc's user avatar
6 votes
1 answer
408 views

Harmonic analysis for a beginner

I am currently dealing with discrete Fourier transform and correlation technique to construct the spectrum of a broad band signal. It's already known that if I have enough observations of the signal, ...
CfourPiO's user avatar
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3 votes
0 answers
48 views

Norm under Gelfand map vs norm under left regular representation on $\ell^p$

Let $G$ be a discrete commutative group. Let $p \in [1,\infty)$ and consider the left regular representation $\lambda : \ell^1(G) \to \mathcal{B}(\ell^p(G))$; that is $\lambda(x)y := x*y$, where $$ (x*...
Leo Sera's user avatar
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2 votes
0 answers
93 views

Anisotropic Calderon-Zygmund decomposition

I am looking for the following version of Calderon-Zygmund decomposition, consider an function $f \in L^1(R^{d+1})$ and cylinders of the form $Q_{R,R^p}$ for some fixed $p \in (0,\infty)$, The ...
Adi's user avatar
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2 votes
0 answers
163 views

How to show that these two functions are $L^\infty$ multipliers?

Suppose there are two multipliers, $$m_1(\xi)=\frac{|\xi|^s}{(1+|\xi|^2)^\frac{s}{2}}$$ and $$m_2(\xi)=\frac{(1+|\xi|^2)^\frac{s}{2}}{1+|\xi|^s}$$ where $s\in (0,\infty)$. My question is: are they $L^\...
Jiawen Zhang's user avatar
1 vote
0 answers
143 views

Almost everywhere convergent Fourier series

Apparently there is a deep theorem stating: Let $f:\mathbb{R}\to \mathbb{C}$ be a function satisfying $f(x)=f(x+2\pi)$ and $\int_0^{2\pi}|f(x)|^2dx<\infty$. Then the Fourier series of $f$ converges ...
Alexander's user avatar
4 votes
2 answers
489 views

A proof of Bernstein's inequality

I'm studying the Meyer's book, "Wavelets and operators", and I'm confused about a proof of Bernstein's inequality at page 47, which is stated below: "The function $\frac{\xi^\beta}{|\xi|...
Jiawen Zhang's user avatar
4 votes
0 answers
68 views

Maximal function estimate for differential quotient of function satisfying $\nabla f \in BMO$

For a function $f \in W^{1,p}(\mathbb R^N)$, it is well-known that there exists a constant $C_N$ (dependent on $N$) such that $$ |f(x)-f(y)| \le C_N|x-y|(\mathcal M|\nabla f|(x) + \mathcal M|\nabla f|(...
user298455's user avatar
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Application of the $\operatorname{BMO}$, $H^1$ duality

Let $f\in \operatorname{BMO}(\partial \Delta)$, then there exists a Carleson measure $\mu$ in $\Delta$ such that $$f(\zeta)-\int_{\Delta}P_{z}(\zeta)d\mu(z)\in L^{\infty}(\partial \Delta),\ \zeta\in\...
Ferry Tau's user avatar
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0 answers
72 views

$|\partial $ as Fourier multiplier

I have the following nonlinear dispersive PDEs $$i \partial_t u- \partial_x^2 u =|\partial_x| |u|^2$$ where $f$ is some nice complex-valued function. I am trying to use the ansatz $u(t,x) = e^{i \...
Mr. Proof's user avatar
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2 votes
1 answer
140 views

Why is the natural map $\hom(A,\mathbb{R}/\mathbb{Z})\to K/A$ an isomorphism, $K/\mathbb{Q}_p$ unramified, $A=\mathcal{O}_K$?

While looking at an analogue of Pontryagin duality for compact Discrete Valuation Rings (DVRs), I came about the observation that generally one should have an isomorphism of $A$-modules $$\hom_{\...
Milo Moses's user avatar
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1 vote
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Holomorphic "quasi-interpolation" of a function sequence

I am interested in some sort of analytic interpolation. A toy version of my problem is as follows. Let $V \subset \mathbb{C}$ be a complex neighborhood of $[0,1]$. Assume there is some bounded ...
Sébastien Loisel's user avatar
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0 answers
99 views

Wavelet decomposition of $C^{k}$-functions on smooth manifolds

Background (compactly supported wavelet decomposition of $\mathbb{R}^n$): Fix compactly supported “mother and father wavelets” $\phi,\psi^{\epsilon}:\mathbb{R}^n\rightarrow \mathbb{R}$ where $\epsilon$...
ABIM's user avatar
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1 vote
1 answer
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Analytic continuation of the Eisenstein series defined over Hecke and Fricke subgroups

It is well known that the (real analytic) Eisenstein series is defined, in the slash notation, as follows $$E_{s}(\tau) = \sum\limits_{\gamma\in\Gamma_{\infty}\backslash\text{SL}(2,\mathbb{Z})}\left.y^...
Spoilt Milk's user avatar
1 vote
0 answers
67 views

Mandelbrot's lacunarity realized by fractal or stochastic field?

It is my understanding that Mandelbrot came up with the notion of lacunarity to classify the homogeneity of 2D functions that only take two distinct values see here. I wonder, does there exist a ...
Guido Li's user avatar
0 votes
0 answers
68 views

Fourier coefficient of close functions

Let $p$ be some prime. Let $\mathbb{Z}_p$ be the cyclic group of order $p$. Let $f, g \colon \mathbb{Z}_p \to \{\pm 1\}$ be two functions. Recall that the Fourier transform is defined as $$ f(x) = \...
Napoleon's user avatar
2 votes
1 answer
257 views

Qualitative difference between "continuous" and "discontinuous" states on $M(G)$

Let $G$ be a locally compact Abelian group (we can think that $G={\mathbb R}$). Let $C_0(G)$ be the space of continuous functions $u:G\to{\mathbb C}$ vanishing at infinity with the usual $\sup$-norm, ...
Sergei Akbarov's user avatar
1 vote
0 answers
131 views

Splitting of a topological vector space (TVS) into an (a) countable sum and (b) direct integral of subspaces

I thought that this would be a simpe question, and placed it here at the Mathematics Stackexchange. Now have to elevate it to Mathoverflow. LANGUAGE TVS = topological vector space. Any subspace of a ...
Michael_1812's user avatar
4 votes
1 answer
286 views

Sharpest version of semiclassical Calderon-Vaillancourt theorem

Let $S$ be the space of symbols defined by $$S:=\{a\in C^{\infty}(T^*\mathbb{R}^d):\forall \alpha,\beta\in\mathbb{Z}^d,\, |\partial_x^{\alpha}\partial_{\xi}^{\beta}a(x,\xi)|\le C_{\alpha\beta}\},$$ ...
Yonah Borns-Weil's user avatar
1 vote
0 answers
128 views

BMO estimates of singular integral operators on torus

I have the following elliptic problem: $$ \Delta u = \operatorname{div}\operatorname{div}S, $$ where $S=(S_{i,j})\colon \mathbb{T}^n\to \mathbb{R}^{n\times n} $ is bounded and $\mathbb{T}^n$ is the $n$...
M_S's user avatar
  • 123
3 votes
1 answer
125 views

Weak convergence of integral averages

Note: This is a refinement of a previous problem. Let $f \in L^1 (\mathbb R)$. Suppose $g_n \in L^1 (\mathbb R)$ are a sequence of positive functions. Define, for each $n$, the function $f_n$ by $$f_n ...
Nate River's user avatar
  • 4,822
6 votes
1 answer
270 views

Convergence of integral averages in $L^1$

Let $f \in L^1 (\mathbb R)$. Suppose $g_n \in L^1 (\mathbb R)$ are a sequence of positive functions. Define, for each $n$, the function $f_n$ by $$f_n (x) := \frac{1}{2g_n (x)} \int_{x - g_n (x)}^{x + ...
Nate River's user avatar
  • 4,822
2 votes
0 answers
214 views

Fourier transform of Dirac delta distribution

Let $f,g$ be Schwartz functions on $\mathbb R^4$, we denote them as $\mathcal S(\mathbb R^4)$, one can then define the transform $V$ mapping $f,g$ to a Schwartz function $\mathcal S(\mathbb R^8)$ $$ V(...
Guido Li's user avatar
2 votes
0 answers
171 views

Illustration of Liouville theorem

In a class, I'll teach the Liouville theorem for harmonic functions with finite Dirichlet integral. What kind of illustrations can I use to elucidate the meaning and proof of the theorem? Note that a ...
user avatar
1 vote
2 answers
638 views

Decay of the Fourier transform of a non-differentiable function

It is well known that if $\varphi$ is a Schwartz function on $\mathbb{R}$ (i.e. smooth and decaying at infinity faster than polynomials), then its Fourier transform decays faster than polynomials. ...
Tony419's user avatar
  • 401
6 votes
0 answers
193 views

Equivalent forms of Fourier restriction conjecture

this question is posted in mathstackexchange, but it seems that no one answers it. Sorry to the administrator if this question is not appropriate on Mathoverflow. I'm reading Pertti Maattila's book ...
Tutukeainie's user avatar
2 votes
0 answers
80 views

Multipole expansion

In Simon's book Harmonic Analysis, example 3.5.12 shows: Fix $x \in \mathbb{R}^{\nu}$ (with $\left.\nu \geq 3\right), x \neq 0$ and define $f$ on $\{y:| y|<| x |\}$ by $$ f(y)=|x-y|^{-(\nu-2)}. $$ ...
Flying ant's user avatar
1 vote
0 answers
60 views

Characterization of elements of Hardy Space

Let $\Omega\subset\mathbb{C}^n$ be a $C^{\infty}$ bounded domain. Let $H^2(\partial\Omega)$ denote the Hardy space, and $S(.,.)$ denote its Szego Kernel. We know that $$ \forall f\in H^2(\partial\...
Naruto's user avatar
  • 63
5 votes
1 answer
283 views

Tensoring with an induced representation: proof question

Let $G$ be a locally compact Hausdorff group and $H$ a closed subgroup of $G$. If $\sigma: H \to B(\mathcal{K}_\sigma)$ is a unitary representation of $G$, we can associate an "induced ...
Andromeda's user avatar
  • 189
3 votes
1 answer
220 views

A sharp estimate for an oscillatory integral with a simple phase

Let $\alpha>1$ not necessarily an integer, and let $\psi:\mathbb{R}\rightarrow \mathbb{R}$ be a smooth function with compact support. Consider the oscillatory integral $$I(\lambda):=\int_{0}^{\...
Medo's user avatar
  • 698
2 votes
0 answers
81 views

Question on a remark in Speh's paper

I am reading Birgit Speh's paper entitled "Unitary representations of Gl(n,R) with nontrivial (g,K)-cohomology" in Invent. Math. 71 (1983), no. 3, 443–465. In Remark 1.2.2.(b), it says that &...
user42804's user avatar
  • 1,091
2 votes
0 answers
78 views

An square root of the multiplicative operator on $\ell^1(\mathbb{Z}_n)$

Let us consider the finite group algebra $\ell^1(\mathbb{Z}_n)$. Let $x=(x_0,\cdots,x_{n-1})$ in $\ell^1(\mathbb{Z}_n)$ and define $$M_x: \ell^1(\mathbb{Z}_n)\to \ell^1(\mathbb{Z}_n) : M_x(a)=a*x$$ ...
ABB's user avatar
  • 3,898
2 votes
0 answers
17 views

Do Szego Kernel in one variable by fixing another variable in a $C^{\infty}$ bounded domain is Bounded?

Let $\Omega\subset\mathbb{C}^n$ be any $C^{\infty}$ bounded domain. Let $ S(.,.)$ denotes the Szego Kenel of Holomorphic Hardy Space $H^2(\partial\Omega)$. Then for $w\in\Omega$ do $S(.,w)$ is a ...
Naruto's user avatar
  • 63
2 votes
1 answer
122 views

Regarding basis of holomorphic Hardy space

Let $\Omega\subset\mathbb{C}^n$ be any $C^{\infty}$ bounded domain and let $H^2(\partial\Omega)$ denotes a holomorphic Hardy space which is a $L^2(\partial\Omega)$ closure of $A^{\infty}(\Omega)(=\...
Naruto's user avatar
  • 63
2 votes
1 answer
236 views

An observation of Nazarov on $\Lambda(p)$-systems

I have been reading this note of F. Nazarov "Summability of large powers of logarithm of classic lacunary series and its simplest consequences (1995)", available here https://users.math.msu....
S117's user avatar
  • 21
1 vote
0 answers
98 views

Parseval-Plancherel identity involving two functions and Riesz transform

Let $J^s = (\mathbb I - \Delta)^{s/2}$, and, $\hat{\mathcal{R}_x u}(\xi)=\frac{-i \xi_1}{|\xi|} \hat{u}(\xi)$, and $u \in H^\infty(\mathbb{T}^2)$, where $u:(\mathbb R, \mathbb{T}^2) \to \mathbb C.$ ...
Mr. Proof's user avatar
  • 159
5 votes
1 answer
184 views

Dense subspaces of the Hardy space $H^1$

Let $H^1$ be the Hardy space on $\mathbb{R}^n$, defined e.g. as the set of $u\in L^1$ such that $Ru\in L^1$, where $R$ is the Riesz transform on $\mathbb{R}^n$. It seems to me that simple functions ...
Piero D'Ancona's user avatar
2 votes
1 answer
191 views

$|\hat\mu(\xi)| \lesssim |\xi|^{-1/2}$ where $\mu$ is $f\mapsto \int_{\mathbb R} \psi(r) \int_{S^{1}} f(rx,r)\, d\sigma(x)\, dr $

I have questions about the proof of Theorem $2.1$ here. The proof is on Pg. $10$. I am trying to work out the $d = 2$ case in particular. $$\mathcal C^d = \{(x_1, \ldots, x_{d+1}): |(x_1, \ldots, x_d)|...
stoic-santiago's user avatar
3 votes
1 answer
212 views

Why does the Fourier algebra $A(G)$ consist precisely of the set of matrix coefficients of the LRR?

This is my first overflow question, so let me apologize in advance if this question is too low level. I was asking the same question in stackexchange, but didn't get an answer; check here for details. ...
jgrk's user avatar
  • 33
3 votes
1 answer
218 views

A kind of vector-valued Littlewood–Paley inequality for arbitrary intervals

The title may be inappropriate, and I apologize for that. I'm writing a reading report on my harmonic analysis course. My topic is the Littlewood-Paley inequality for arbitrary intervals, which was ...
Feng's user avatar
  • 517
2 votes
1 answer
296 views

Difference equation satisfied by discrete harmonic functions on square lattice

A function $f:\mathbb Z^2 \rightarrow \mathbb R$ is said to be discrete harmonic if it satisfies the discrete Laplacian equation $$ \Delta f(m,n) = f(m+1,n)+f(m-1,n)+f(m,n+1)+f(m,n-1)-4f(m,n) = 0~. $$ ...
Pranay Gorantla's user avatar
2 votes
1 answer
215 views

Extension of a Szegő Kernel to the boundary

Let $\Omega\subset\mathbb{C}^n$ be any smooth bounded pseudoconvex domain. Let $S$ denote the Szegő kernel of $\Omega$. Recall: the Szegő kernel is a kernel of the Szegő projection $P: L^{2}(\partial\...
Naruto's user avatar
  • 63
2 votes
0 answers
83 views

Maximal function to high power

Consider the following maximal function : in dimension $n$ consider $B(0,1)\subset \mathbb{R}^n$ the unit ball, if $f\in L^1(B(0,1))$, $\alpha\geq 0$ : $$ M_\alpha f : x \mapsto \sup \left\{ \frac{1}{...
Dorian's user avatar
  • 331
3 votes
1 answer
150 views

Urysohn's lemma for Bochner functions?

Let $G$ be a locally compact Hausdorff group. In the proof of theorem 4.32 of Folland's book "A course in abstract harmonic analysis", the following result is used: If $U$ is an open ...
Andromeda's user avatar
  • 189
5 votes
1 answer
262 views

de Rham theorem for tempered distributions

I am wondering if the following statement holds. If $u\in \mathscr{S}'$ satisfies $\left< u,\Phi\right>=0$ for all $\Phi \in \mathscr{S}$ with $\mathrm{div}\, \Phi=0$, then there exists $p\in \...
Will Kwon's user avatar
  • 323
4 votes
0 answers
69 views

Maximal function in Orlicz space

Consider the maximal operator defined for a function $f\in L^1_{loc}$: $$ Mf : x\mapsto \sup_{r>0} \frac{1}{|B(x,r)|} \int\limits_{B(x,r)} f. $$ It is well know that $M : L^1 \to L^{(1,\infty)}$ ...
Dorian's user avatar
  • 331
1 vote
2 answers
152 views

Is $\int_{\mathbb{R}} \int_{\mathbb{R}^n} \alpha w(t) e(\alpha (a_1t_1 + \dotsb + a_n t_n)) dt\,d \alpha = 0$?

Let $a_i$ be a nonzero real number for each $1 \leq i \leq n$. $w$ a smooth nonnegative with compact support. I would like to understand the following integral. $$ I = \int_{\mathbb{R}} \int_{\mathbb{...
Johnny T.'s user avatar
  • 3,547
2 votes
0 answers
59 views

Does periodic pattern arise in syndetic pattern

We wonder for two "large" sets $I,J\subseteq \omega$, if $(J-J)\cap I=\emptyset$, then it must be due to certain periodic pattern. We say $I\subseteq \omega$ is $t$-syndetic iff for every $n\...
Jiayi Liu's user avatar
  • 909
5 votes
1 answer
266 views

About weak integrals: Appendix of Folland's book "A course in abstract harmonic analysis"

Consider the following fragment from Folland's book "A course in abstract harmonic analysis": All integrals are here to interpreted in the weak sense (see p285 in Folland's book). Why is ...
Andromeda's user avatar
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