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.

Filter by
Sorted by
Tagged with
3 votes
1 answer
108 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 ...
user avatar
  • 261
5 votes
1 answer
203 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 \...
user avatar
  • 303
3 votes
0 answers
39 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 $M : L^1 \to L^{(1,\infty)}$ ...
user avatar
  • 63
1 vote
2 answers
134 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{...
user avatar
  • 2,463
2 votes
0 answers
46 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\...
user avatar
  • 909
5 votes
1 answer
196 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 ...
user avatar
  • 261
-1 votes
0 answers
105 views

Eigenvalues of the Casimir operator and the space $L^2(E)$

I have read something about harmonic analysis on a homogeneous vector bundle in a very interesting paper Simultaneous Linearization of Diffeomorphisms of Isotropic Manifolds by J. DeWitt in my M.Sc. ...
user avatar
4 votes
0 answers
143 views

Hodge theory in higher eigen-spaces?

Hodge theory for elliptic complexes $E$ identifies the space of harmonic sections with cohomology $$\mathcal{H}(E) \simeq H(E).$$ A classical example with differential forms ($E = (\Omega,d)$) ...
user avatar
  • 4,005
1 vote
0 answers
72 views

Irreducible unitary representations of $\mathrm{SL}(n,\mathbb R)$ from those of $\mathrm{GL}(n,\mathbb R)$

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$In the case of a non-Archimedean local field $\mathbb F$, one may reduce the representation theory of $\SL(n,\mathbb F)$ to that of $\GL(n,\...
user avatar
  • 1,031
3 votes
1 answer
52 views

Analytic approximation of the step function in $L^p$ norm

Motivation: Euler-Maclaurin formula uses calculus to estimate discrete sums. I wonder what one can do by reverse engineering. A concrete problem I ran into is the following. Question: Let $\chi: \...
user avatar
  • 4,005
1 vote
0 answers
30 views

Riesz transform of constant function

My one-line question would be, what is the Riesz transform of the constant function, identically equal to 1 on $\mathbb{R}^2$? But more fundamentally, my question stems from some confusion about the ...
user avatar
  • 267
2 votes
0 answers
137 views

An oscillatory integral

Let $s>0, v\in \mathbb{R}^d, w\in \mathbb{R}, |w|\leq 1$. Pick a cut-off function $B(0,1)\prec \eta \prec B(0,2)$ and a large real number $N$. Do we have the following type of estimates? \begin{...
user avatar
2 votes
0 answers
141 views

A closed ideal in $L^1(T)$

Let $\mathbb{T}$ be the unit circle and consider the convolution group algebra $L^1(\mathbb{T})$. Let $I_n$ be the closed ideal generated by the polynomial $p_n(z)=z^n-1$ in $L^1(\mathbb{T})$. Let $I=...
user avatar
  • 3,343
4 votes
1 answer
167 views

Integral of $\ln(1/|f|)$ for $f$ bandlimited

I came across the following assertion: if $f\in PW_\infty([-a,a])$, i.e. the Bernstein space of functions in $L^\infty(\mathbb{R})$ which are the Fourier transform of a distribution supported on $[-a,...
user avatar
  • 215
2 votes
1 answer
129 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(\...
user avatar
  • 49
2 votes
1 answer
126 views

Exponential integrability of a sum of approximations of disjoint intervals characteristic functions

Let $I=[0,1]$ be the unit segment, and let $(I_n)_{1\leq n \leq N}$ be $N$ almost disjoint sub-intervals $I_n=[t_n-\delta_n,t_n+\delta_n]$ of $I$ (that is, their interior are disjoint). Let $\chi(x)=\...
user avatar
  • 367
3 votes
2 answers
182 views

A Sobolev embedding theorem for functions on spheres

$L^2(\mathbb{S}^{d-1})$ is embedded in $H^{-s}(\mathbb{R}^d)$ with $s>\frac{1}{2}$, which means for $f\in L^2(\mathbb{S}^{d-1})$, the following holds: $$\DeclareMathOperator{\Dm}{\operatorname{d}\!}...
user avatar
3 votes
0 answers
68 views

Unitary with entries $(i,j)$ only on equidistant lattice points $\|i-j\|^2 = c^2 \in \mathbb{N}$

My research needs help in finding examples of unitary matrices $U$ which have entries \begin{align} U_{ij} = \begin{cases} \alpha_{ij}, \ \text{ if } \|i-j\|^2 = c^2 \\ 0 , \text{ otherwise} \end{...
user avatar
  • 41
7 votes
2 answers
648 views

Estimates about prime numbers: a lemma in Bourgain's article

For $n\in \mathbb{N}$ with prime decomposition $n=p_1^{r_1}\cdots p_k^{r_k},p_i\neq p_j$, let $A=\{p_1,\cdots,p_k\}$; then the following holds: \begin{equation} |\{q\in \mathbb{N},q<Q: \text{all ...
user avatar
0 votes
0 answers
36 views

Description of Sobolev functions with rapidly decaying wavelet expansion

Consider the Sobolev space $W^{k,p}((0,\infty))$ and fix a wavelet basis $\{\phi_i\}_{i=0}^{\infty}$ of for it (where $1\leq p<\infty$ and $0<k<\infty$). Since $\{\phi_i\}_{i=0}^{\infty}$ is ...
user avatar
  • 55
0 votes
1 answer
109 views

Finite Hindman theorem

Consider the following finite version Hindman theorem: For every sufficiently large $N\in\omega$ and 2-partition of $N=N_0\cup N_1$, there are $i<2,a,b,c\in N_i$ such that $a+b=c$. The only proof I ...
user avatar
  • 909
7 votes
0 answers
117 views

The relation between Wolf's and Teichmüller's parametrization of the Teichmüller space

Let $\mathcal{T}_g$ be the Teichmüller space of Riemannian surface structures on an oriented 2-dimensional manifold of genus $g$. Fix a point $S \in \mathcal{T}_g$. There are two different ways to ...
user avatar
1 vote
0 answers
43 views

How to show the solution map of NLS is not smooth?

Let $u(\delta, t)$ satisfy $$iu_t +\Delta u+ |u|^{2k}u=0, \quad u(0)=\delta v_0$$ Note that the mapping: $$\delta v_0\mapsto u(\delta, t)= S(t)(\delta v_0)-i\int_0^tS(t-\tau)(|u|^{2k}u)(\tau)d\tau $$ ...
user avatar
4 votes
1 answer
163 views

Maximal ergodic inequality

A map $f: X \to X$ preserves an ergodic probability $\mu$, i.e., $\mu \circ f^{-1}=\mu$ and for any $\phi: X \to \mathbb{R}$ with $\int \phi d\mu=0$, $$\frac{1}{n} \sum_{i \le n} \phi \circ f^i \to 0 \...
user avatar
  • 43
0 votes
0 answers
42 views

Euclidean log Sobolev inequality for fractional Laplacian

I am interested in this work of Cotsilois and Tavoularis, which claims the following Euclidean log Sobolev inequality for the fractional Laplacian, generalizing the classical Gross-Stam log Sobolev ...
user avatar
12 votes
0 answers
193 views

Pointwise convergence of trigonometric series

$f$ is said to have trigonometric expansion if some series $\sum_{n\in\mathbb{Z}}c_ne^{inx}$ converges pointwise to $f(x)$. On the second page of the article Trigonometric series and set theory, ...
user avatar
  • 121
10 votes
2 answers
536 views

Existence of a strongly continuous topologically irreducible representation of a compact group on an infinite dimensional Banach space?

Does there exists a triple $(G, X, \pi)$, where $G$ is a compact group, $X$ an infinite dimensional Banach space over $\mathbf{C}$, and $\pi : G \to B(X)$ a strongly continuous representation of $G$, ...
user avatar
  • 814
3 votes
1 answer
549 views

Equivalent action of convolution of directional derivative

I have asked this question a while back on StackExchange but have not received any answer/comment. I received a suggestion to post the same question in here which is more research oriented. Let $k*f(x)...
user avatar
  • 49
1 vote
1 answer
113 views

Equality of two subharmonic functions

Let $u\leq v$ be two locally bounded subharmonic functions in a domain in $\mathbb{R}^n$. Assume that $u=v$ on a dense subset. Is it true that $u=v$ everywhere?
user avatar
  • 19k
1 vote
0 answers
242 views

Stein's book on harmonic analysis

My background : I am a Math PhD student. I will most probably work in harmonic analysis on Euclidean spaces. I am a fan of Folland's Real analysis and I have thoroughly studied first 8 chapters of ...
user avatar
9 votes
2 answers
1k views

Density of restrictions of harmonic functions inside a ball

Let $B$ be the closed unit ball in $\mathbb R^3$ centered at the origin and let $U= \{x\in \mathbb R^3\,:\, \frac{1}{2}\leq |x| \leq 1\}.$ Let $$ S_U= \{u \in C^{\infty}(U)\,:\, \Delta u =0 \quad\text{...
user avatar
  • 2,858
1 vote
0 answers
54 views

A problem arising from Wiener-Levy theorem on the real line

Theorem (Wiener-Levy). Let $A(\mathbb{T})$ be the Fourier-algebra on the unit circle $\mathbb{T}$. Let $f$ be in $A(\mathbb{T})$ and suppose that $F$ is an analytic function on the range of $f$. Then $...
user avatar
  • 3,343
1 vote
0 answers
33 views

Stability of Hajłasz-Sobolev class under post-composition

Informally: When is a Sobolev function, post-composed by a vector-valued function still Sobolev? Assumptions/Setup Let $(X,d_X,m_X)$ and $(Y,d_Y,m_Y)$ be complete and separable metric measure spaces; ...
user avatar
0 votes
0 answers
56 views

Extracting the point mass measure of some type of positive measures

Let us consider the measure algebra $M(\mathbb{R})$ consisting of all Radon measures on the reals. Let $\delta_0$ be the point mass measure concentrated on 0, which is also the multiplicative ...
user avatar
  • 3,343
0 votes
1 answer
62 views

An equation in the convolution measure algebra on reals

Let us consider the measure algebra $M(\mathbb{R})$ consisting of all Radon measures on reals. Let $\mu$ be a Radon measure in $M(\mathbb{R})$ and $\delta_0$ be the point mass measure concentrated on ...
user avatar
  • 3,343
3 votes
0 answers
94 views

Is the cone of positive elements in $L^1(G)$ norm closed?

Let's consider $L^1(G)$, the Banach $*$- algebra of all Lebesgue integrable functions on the locally compact group $G$. Put $L^1(G)_+$ by the cone of positive elements given by $\{\sum_1^n f_i^**f_i: ...
user avatar
  • 3,343
1 vote
0 answers
42 views

Real life applications of distributions through models or simulations [closed]

What are the areas we can apply distributions in classical harmonic analysis? I don't mean probability distributions but distributions that are continuous linear functionals on the space of test ...
user avatar
  • 61
0 votes
0 answers
47 views

How can C2 double-helices, produce C3 rotational symmetry?

I am trying to solve the structure of a helical protein complex using a software for maximum a posteriori refinement of (multiple) 3D reconstructions (Relion). When searching for helical symmetry, the ...
user avatar
6 votes
1 answer
168 views

Oscillatory integrals with a decaying factor in the integrand

Lemma 4.5 of Titchmarsh's book The Theory of the Riemann Zeta function says (slightly rephrased): Let $F$ be a twice differentiable real function such that $ F''(x) \geq r > 0$ for all $x$ in $[a,b]...
user avatar
  • 63
5 votes
1 answer
344 views

Unifying two definitions of $L^\infty$

Let $X$ be a locally compact Hausdorff space and $\mu$ a Radon measure on $X$. Definitions: A subset $E\subseteq X$ is called locally Borel if $F \cap E$ is Borel for every Borel set $F\subseteq X$ ...
user avatar
  • 261
1 vote
0 answers
77 views

Inequality on the dual space of $H^s$

Does there exist a theorem that allow us to say that, if we have an estimate on the Sobolev space $H^s\,,\, s\geq 0$ then we can deduce an estimate on the dual space $H^{-s}$ ? For instance, assume ...
user avatar
  • 63
4 votes
1 answer
193 views

Probability measure on the boolean cube with small support and small Fourier transform

[Edit: added details on the Reed-Muller codes] Are there explicit (non random) constructions of probability measures on $D_N = \{0,1\}^N$ with support of size $O(N)$ and with all nontrivial Fourier ...
user avatar
2 votes
0 answers
49 views

Orthogonal basis of martingale-Hardy space

Let $(X,\mathcal{B}(X),(\mathcal{A}_n)_{n=1}^N,\mu)$ be a filtered probability space with $\mathcal{A}_N=\mathcal{B}(X)$ and let $H$ be the space of $\mathcal{A}_{\cdot}$-adapted martingales $m_{\cdot}...
user avatar
3 votes
0 answers
107 views

Erdős–Turán inequality for complex numbers

Consider the following set of complex numbers in the upper half plane: $$\{ic_n \pm \gamma_n: 0 \leq n \leq N, \hspace{1 mm} c_0=\gamma_0=0, c_n,\hspace{1 mm} \gamma_n>0\}.$$ Assume that this set ...
user avatar
3 votes
1 answer
128 views

Fourier multipliers and transference on cyclic groups

It seems to be a commonplace in harmonic analysis that if some operator (say, Fourier multiplier) is bounded on $L^p(\mathbb{R}^n)$ then by transference the similar operator is also bounded on $L^p(\...
user avatar
2 votes
0 answers
106 views

Is there a characterization of tempered functions whose Fourier transforms are also tempered functions?

A tempered function on $\mathbb{R}^n$ is a locally integrable function that is tempered as a distribution, i.e. $L^1_{loc}\cap\mathcal{S}'$ is the space of tempered functions. This MSE question asked ...
user avatar
3 votes
0 answers
116 views

On the best constant for Carleson's embedding theorem

In "Interpolations by bounded analytic functions and the corona problem", Carleson introduced Carleson measures (for Hardy spaces) and proved the famous embedding theorem according to which $...
user avatar
2 votes
0 answers
114 views

Non-commutative harmonic analysis on the discrete Heisenberg group

Question: Is there a linear map $\mathcal F$ from the Hilbert space of $\ell^2$ functions on the discrete Heisenberg group to some Hilbert space of functions $ L^2(\bigcup \{\Omega_n\}) $, such that:...
user avatar
  • 7,982
6 votes
0 answers
150 views

Regularity of $|u|^{\alpha}$ when $u$ is Schwartz

Let $0<\alpha<1$. Let $D_x^{\alpha}$ denote the Fourier multiplier given by $\xi\to |\xi|^{\alpha}$. Suppose $u:\mathbb{R}^d\to\mathbb{C}$ is Schwartz (or even just smooth with compact support). ...
user avatar
3 votes
0 answers
103 views

Regularity of solution to Laplacian equation with Neumann data on Lipschitz domain

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$ and let $u\in H^1(\Omega)$ be a weak solution to \begin{equation} \begin{cases} -\Delta u=0 \quad &\mbox{in $\Omega$}\\ \frac{\partial ...
user avatar
  • 1,040

1
2 3 4 5
24