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 ...

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238 views

harmonic extension of a curve by different parametrization

Let us consider a curve $\gamma :S^1 \rightarrow \mathbb{R}^3$ (or even a planar convex one if it simplifies). Then I look to the harmonic extension to the disc $h:\mathbb{D}\rightarrow \mathbb{R}^3$ (...
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3answers
332 views

Fundamental solution of Discrete Laplace in the plane

We consider a discretization of the Laplace operator on $\mathbb Z^2$, https://en.wikipedia.org/wiki/Discrete_Laplace_operator Then, it is natural to consider its fundamental solution $u$, i.e. $|u(x)...
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votes
1answer
80 views

Hilbert transform on boundary value of analytic bounded functions

I am considering the boundary values of a bounded holomorphic functions. Suppose $w$ is a bounded holomorphic function in upper half plane, with continuous and bounded boundary value $f$ on real axis. ...
9
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3answers
563 views

Historical developement of analysis and partial differential equations (especially in the 20th century)

Q: Is there a set of some comprehensive surveys or monographs describing (in technical detail) the historical development of the various subareas of analysis and partial differential equations? ...
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108 views

A kaleidoscopic coloring of the plane

Is there a partition $\mathbb R^2=A\sqcup B$ of the Euclidean plane into two Lebesgue measurable sets such that for any disk $D$ of the unit radius we get $\lambda(A\cap D)=\lambda(B\cap D)=\frac12\...
9
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1answer
315 views

Pontryagin dual of the surreal numbers?

Has any work been done on the Pontryagin dual of the surreal numbers (suitably topologized)? I have not been able to find anything and am not sure if this is still unknown. Alternatively, has this ...
8
votes
2answers
529 views

Can exponential sums be small on a whole interval?

This is almost certainly routine to an analyst, so forgive me in advance. Let $\alpha_i\in \mathbb{R}$. Consider the functional $$\varphi: L^1[0.9A,A]\to \mathbb{C}$$ via $$f\mapsto \sum_i \hat{f}(\...
3
votes
2answers
170 views

Name for an orthogonal decomposition of $L^2 (\mathbb{R}^2; \mathbb{C})$

The space $L^2 (\mathbb{R}^2; \mathbb{C})$ can be decomposed as $$ L^2 (\mathbb{R}^2; \mathbb{C}) = \bigoplus_{k \in \mathbb{Z}} L^2_k (\mathbb{R}^2; \mathbb{C}), $$ where $$ L^2_k (\mathbb{R}^2; \...
20
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1answer
530 views

Bounding Schur symmetric polynomials on the unit circle

Recall the Schur polynomial in $n$ variables, indexed by the partition $\lambda$, with $\ell(\lambda) \leq n$, is given by \begin{equation} s_\lambda(x_1,\ldots, x_n) = a_{\lambda + \delta}(x_1, \...
2
votes
3answers
187 views

Equivalence of Harmonic Maps and Conformal Maps on Genus-0 closed surfaces

By the uniformization theorem, for every genus-0 closed surface $\mathcal{M}\subset\mathbb{R}^3$, there is a conformal map $f:\mathcal{M}\rightarrow \mathbb{S}^2$. Furthermore consider the Dirichlet ...
1
vote
1answer
123 views

Spherical decreasing rearrangement on the sphere

On $\mathbb{R}^n$, we have the concept of spherically decreasing rearragement of a function, which means, given a function $f$, one can design a radial and decreasing function $f^*$ such that $\Vert f^...
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111 views

Looking for some “nontrivial” examples of pseudodifferential operators/symbols

I'm reading up on $\Psi DO$'s and trying to find some examples of symbols that are not quite so trivial. Obviously, the first example of a symbol that most people talk about is just a polynomial in $\...
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vote
1answer
63 views

Negativity of a quadratic form on $L^2(M)$

Let $M$ be a compact Riemannian manifold and $V=L^2(M)$. Let $\Delta$ be the negative-definite Laplacian. Let $f \in V$ and $x \in M$ be arbitrary, but fixed. Is it true that ${\rm Re} \ (\Delta f) (...
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215 views

Wavelet-like Schauder basis for standard spaces of test functions?

The Schwartz space of test functions $\mathcal{S}(\mathbb{R})$ is isomorphic to $\mathfrak{s}$ the space of sequences of real numbers with faster than power-like decay. Likewise, the space $\mathcal{D}...
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56 views

Dimension of the set of the polynomial growth harmonic function on the hyperbolic plane

We consider the hyperbolic plane and the harmonic function there. Pick any point $p$. Let $H_n, n \in\mathbb N$ be the set of the harmonic functions $f$ such that $|f(x)|\leq c(1+ d(x,p))^n$. What is ...
2
votes
1answer
188 views

Hardy space, Lebesgue space for $p<1$,

We denote $\mathcal D'(\mathbb R^n)$ the space of distributions, and $\mathcal D(\mathbb R^n)$ the space of smooth, compactly supported functions. Let $\rho\in \mathcal D'(\mathbb R^n)$ such that ...
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42 views

The Best Korn's constant for bounded deformation

I am studying the following version of Korn's inequality. For $u\in BD(\Omega)$, $BD$ denotes the bounded deformation space, we have, there exists a $r(u)\in \operatorname{ker}\mathcal E$ such that $$...
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1answer
95 views

One question about the tensor product of $L^1(G)$ and a Banach space $A$

We know that the tensor product of $L^1(G)$ and a Banach space $A$ is isometric to $L^1(G, A)$, the space of all Bochner-integrable $A$-valued functions on a locally compact group $G$. I am looking ...
2
votes
1answer
82 views

Rajchman measures via strong mixing systems

A Rajchman measure on the unit circle $\mathbb{T}$ is a Borel probability measure $\mu$ with $\lim_{n\to\infty}\hat{\mu}(n)=0$. Where $\hat{\mu}(n)=\mu(z^n)$ for $n\in\mathbb{Z}$ are Fourier ...
2
votes
2answers
128 views

The convolution between weighted $L^1$ space and normal $L^1$ space

Let $\omega\in L^1_{\text{loc}}(\mathbb R^N$) be given. We assume that $\omega\geq 1$, l.s.c, and satisfies, for a constant $C>0$, $$ \frac{1}{|B(x,r)|}\int_{B(x,r)}\omega(y)dy\leq C\omega(x) $$ ...
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0answers
50 views

Reference request - Compact embedding of intermediate space

Given two Banach spaces $X_0$ and $X_1$ with norms $\|\cdot\|_0$ and $\|\cdot\|_1$, respectively, such that $X_0\subset X_1$ and $X_0\hookrightarrow X_1$, i.e., $X_0$ is continuous embedded in $X_1$. ...
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0answers
94 views

Why a cone/parabolic set for the nontangential maximal function?

Suppose $f\in L^p(\mathbb{R}^d)$. Then the Dirichlet BVP for the Laplace equation $\Delta u = 0$ in the upper-half plane $\mathbb{R}^d\times\mathbb{R}_{>0}$ with boundary value $f$ can be solved by ...
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298 views

Localization arguments in the paper 'the proof of $l^2$ decoupling conjecture'

I am currently reading Jean Bourgain and Ciprian Demeter's 2015 paper The proof of the $l^2$ decoupling conjecture and would appreciate some help in understanding localization argument used in that ...
4
votes
1answer
313 views

What is the importance of convergence of variation of Fourier reconstruction to that of variation of the function?

Let $f$ be a periodic function of bounded variation which jumps at a point $x_0\in\mathbb{R}$. Let $S_{N}[f]$ denote the partial Fourier sum of $f$ and let $C_{N}[f]$ denote the Cesaro partial sum. It ...
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90 views

Ring of SO(n)-invariant differential operators on M_n,m

I'm reading through Stephen Gelbart's paper "A Theory of Stiefel Harmonics." (http://www.ams.org/journals/tran/1974-192-00/S0002-9947-1974-0425519-8/). There comes a point in the paper (Lemma 2.8) ...
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1answer
254 views

The $L^2\times L^2\to L^2$ norm of the bilinear multiplier operator

Consider a general bilinear multiplier operator: $$ T(f,g)(n)=\int_{\Pi}\int_{\Pi}\hat{f}(\xi)\hat{g}(\eta)e^{2\pi i(\xi+\eta)n}m(\xi,\eta)d\xi d\eta, $$ where $\Pi$ is the torus, $n\in\mathbb{Z}$, $m$...
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0answers
101 views

What does the Plancherel theorem say about positive-definite distributions?

I'm trying to understand the answer to this MO question: Bochner's theorem for measures of positive type, which suggests a relationship between Bochner's theorem and the Plancherel theorem. The ...
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0answers
153 views

Generalization of Pitt's theorem

Pitt's theorem (Pitt 1937), states that the one-dimensional Fourier tranform is well defined and continuous between the weighted spaces $L^p(\mathbb{R},|x|^{bp}dx)$ and $L^q(\mathbb{R},|x|^{\beta q}dx)...
0
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1answer
124 views

Reproducing Kernel of a RKHS of continuous functions may not be continuous in two variables together

Let $\mathcal{K}$ be a Hilbert Space of continuous functions on some topological space, where point evaluations are continuous linear functional on $\mathcal{K}$. That is $\mathcal{K}$ is RKHS, ...
4
votes
2answers
262 views

Equivalent Norms on Sobolev Spaces

When $k$ is a positive integer and $1<p<\infty$, we know that there is some $C>0$ such that for all $u\in W^{k,p}\left(\mathbb{R}^{N}\right) :$ $$ \left\Vert \left( -I+\Delta\right) ^{\...
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0answers
71 views

Relating the R-transform in free probability to noncommutative group representations

In traditional (commutative) probability theory, sums of random variables correspond to convolutions of distribution functions, which plays well with the Fourier Transform. In free (noncommutative) ...
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votes
2answers
362 views

Can phase significantly concentrate a function's spectrum?

Let $F$ denote the Fourier transform over some group. What is known about the following quantity? $$\gamma:=\inf_{x\neq 0}\frac{\|Fx\|_1}{\|F|x|\|_1}$$ Here, $|x|$ denotes the pointwise absolute ...
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vote
2answers
142 views

Are spherical harmonics uniformly bounded?

The spherical harmonics are given by $$Y^m_l(\phi,\theta):=N^m_l e^{im\theta}P^m_l(\cos \phi) $$ where $P^m_l$ are the associated Legendre Polynomials and $N^m_l$ is the normalisation. From ...
5
votes
1answer
377 views

Monge–Ampère with drift

Let $I\subseteq \mathbb{R}$ be an interval. Let smooth $M(x,y):I\times(0,\infty) \to \mathbb{R}$ satisfies PDE: $$ M_{xx}M_{yy}-M_{xy}^{2}+\frac{M_{y}M_{yy}}{y}=0. $$ My question is to describe/...
2
votes
1answer
196 views

Spherical harmonics and ellipticity of the Laplacian

Let us consider the sphere $S^n$ and the Laplacian $-\Delta$ on it. Let $L^2(S^n) = \bigoplus_k V_k$, where $V_k$ represents the eigenspace of the Laplacian with eigenvalue $k(k + n - 1)$. We know ...
5
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0answers
165 views

Does there exist a smooth version of Cohen's factorization theorem?

A famous factorization theorem of Cohen states that for any locally compact group $G$, $$L^1(G)=L^1(G)*L^1(G).$$ I want to know if analogous results exist for the class of smooth functions when $G$ is ...
2
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0answers
129 views

Complex sum of squares of vector fields (hypoelliptic operators)

Consider a compact $M$ of dimension $n$. Consider real smooth vector fields $X_0, X_1, X_2,..., X_n$ on $M$ and consider the differential operator $\mathcal{L}_1 = \sum_{i = 1}^n X_i^2 + X_0.$ Now, by ...
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votes
0answers
68 views

Fourier coefficients of positive polynomials

Let $p(x) \geq 0$ be a positive polynomial on the hypersphere ($x \in S^{n-1}$) satisfying $\int_{S^{n-1}} p(x) = 1$. Writing $p(x) = \sum_{j=0}^s p_j(x)$ where $p_j(x) = \sum_m p_{jm} s_{jm}(x)$ with ...
8
votes
2answers
782 views

Is $f(x,y)=\sum_{n\in\mathbb{Z}\backslash\{0\}}\frac{1}{n}e^{2\pi i(xn+yn^2)}$ essentially bounded?

Let $$f(x,y)=\sum_{n\in\mathbb{Z}\backslash\{0\}}\frac{1}{n}e^{2\pi i(xn+yn^2)} $$ Is it true that $\|f\|_{L^{\infty}(\mathbb{R}^2)}<\infty$? i.e. is $f$ essentially bounded?
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3k views

Eliminating Gibbs phenomenon, and approximating with jumping functions in Fourier Analysis : An attempt and a question in this regard

This problem seems like a nightmare to me. I tried to expand $K_{\omega}^f(t)$, but I am clueless of getting some kind of a closed form or some kernel like structure. If I try to take derivative, ...
2
votes
1answer
213 views

$BMO$-property via a John-Nirenberg type estimate?

Let $\Omega \subset \mathbb R^d, d\ge 2$, be bounded and denote a ball in $\mathbb R^d$ by $B$. Denote also $$ f_B:= \frac1{|B|}\int_B f \, dx. $$ Suppose $f \in L_{\rm loc}^p(\Omega)$ for all $1<p&...
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0answers
81 views

Does the Total variation of the Fourier partial sum of a bv function with jumps converge to TV of the function as $N\to\infty$

Does the total variation of the Fourier partial sum of a piecewise continuous bv function converge to the total variation of the function as $N\to\infty$. To explain briefly, Let $f$ be a periodic ...
5
votes
1answer
127 views

The (Hecke) double coset von Neumann algebra

It it well-known in the von Neumann algebra theory that for $\Gamma$ a non-trivial countable group, the von Neumann algebra $L(\Gamma)$ generating by $\Gamma$ acting by left multiplication on $l^2(\...
2
votes
1answer
194 views

an analogue of Littlewood-Paley-Rubio de Francia theory

For any function $f$ defined on the set of integer $\mathbb{Z}$, we define its Fourier transform as the following periodic function: $$ \mathbb{F}f(\xi)=\sum_{n\in\mathbb{Z}}f(n)e^{-2\pi i n\xi} $$ ...
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vote
0answers
117 views

Technical question about a Fourier transform

I would like to know if there is an explicit expression for the Fourier transform of the following function: $$f(x)=\mathbb{1}_{(0,\infty)}e^{-x-ix^2},$$ or to know where I can find some techniques to ...
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0answers
48 views

Non interacting complex unit

How to work with two non interacting complex units say i and j. These two imaginary complex unit represent different quantities. For example i is for periodicity in theta and j is for frequency or ...
7
votes
1answer
335 views

Dominated convergence to characteristic function

Let $\phi_m(x):=\chi_{[0,1]} * \chi_{[0,1]} *...* \chi_{[0,1]}$ be the m -times convolution (so $m+1$ characteristic functions are involved). Then the Fourier transform of this function is given by $...
7
votes
1answer
212 views

Poincaré inequality for curl-integrable functions

Let $B=B(r)$ denote a ball of radius $r$ in $\Omega \subset \mathbb R^d$ and $$ u_B := \frac1{|B|}\int_B u \, dx. $$ The standard Sobolev-Poincaré inequality states that if $u \in W^{1,p}(\Omega)$, ...
5
votes
1answer
173 views

application of factorization theorem

Young's inequlity tells us that $L^{1}(\mathbb R)\ast L^{p}(\mathbb R) \subset L^{p}(\mathbb R)$ with norm inequality $$\|f\ast g\|_{L^{p}} \leq \|f\|_{L^1}\|g\|_{L^p};$$ and of course this ...
1
vote
1answer
67 views

Limiting absorption principle

I would like to know if there is a book (or a paper) which can give me an introduction to LAP. I tried to read some papers by myself, but I don't feel comfortable. I think that I need the basic ideas ...