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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0answers
23 views

Functions arising as STFT with band-limited window of a compactly supported function

Suppose that $f \in L^2(\mathbb R)$ has support in a compact set $[-A,A]$ and $g \in L^2(\mathbb R)$ has compact support in the frequency domain, say $\mathrm{supp} \, \hat g \subseteq [-B,B]$ where $\...
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52 views

Uniform distribution in a cancelling exponential sum

I have a exponential sum with the following cancellation property: For $0<c_n, \gamma_n<1$ $$f(t):= 1+ \sum_{n=1}^{M} e^{-c_n t\pm i\gamma_n t} = O(\frac{1}{e^{t/4}}),$$ when $t \in [A, B].$ ...
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59 views

Half-integer Fourier transform

Suppose we want to carry out the (generalized) Fourier transform of a function defined in the domain $\mathbb{T}\times\tfrac12\mathbb{Z}$, i.e. dependent on the arguments $\phi\in\left(-\pi,+\pi\right]...
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129 views

Spherical harmonics, $\frak{sl}_2$, and algebra gradings

Let $S^2$ be the usual $2$-sphere considered as the quotient $S^3/S^1$, and denote by $\operatorname{Pol}(S^2)$ the algebra of polynomial functions on $S^2$. We can decompose $\operatorname{Pol}(S^2)$ ...
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180 views

Spherical harmonic expansion of a power function

Let $f$ be an even continuous function on the sphere $S^{n-1}$. Find a relation of the spherical harmonic expansion between coefficients of $f^n$ and those of $f$.
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1answer
143 views

Understanding the regular representation of an LCA group as a 'direct integral'

The reference for what I'm asking is page $107$ from Folland's harmonic analysis. $G$ is a locally compact abelian group with dual $\hat{G}$. Let $H$ denote the Hilbert space $L^2(G)$. I'm trying to ...
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1answer
200 views

Generalization of Bernstein’s inequality

I'm using Muscalu and Schlag's textbook to study harmonic analysis and I encountered the following claim: Given some function $f \in \mathcal{S}(\mathbb{R}^{d})$, where $\mathcal{S}(\mathbb{R}^{d})$ ...
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99 views

$L^2$ convergence of a particular function

I encounter the following problem when I study harmonic analysis by myself: Given a function $f \in L^2([0,1])$. Let's fix some irrational number $\omega$. For any $N \in \mathbb{Z}^{+}$, let's define ...
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90 views

$L^p$ estimate of a multiplier operator

I'm studying harmonic analysis by myself and I encountered the following claim about multipliers: consider a sequence of complex numbers $\{m_{n}\}_{n \in \mathbb{Z}}$ that satisfies: $$\sum_{n \in \...
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1answer
78 views

Is the Besov space $B_{\infty,1}^0(\mathbb{R}^d)$ a multiplication algebra?

Let $s\in\mathbb{R}$ and $1\leq p,q\leq\infty$. Consider the Besov scale of spaces $B_{p,q}^s(\mathbb{R}^d)$ defined by the norm $$\|f\|_{B_{p,q}^s} := (\sum_{j=0}^\infty \|P_{j} f\|_{L^p}^q)^{1/q},$$ ...
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1answer
245 views

Haar measure coming from Pontryagin duality v/s Fourier inversion

Not research but advertising this question from mse in case someone wants to answer. I'm struggling with some bookkeeping associated with the Pontryagin duality theorem. I'm thinking about the first ...
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83 views

Weak-type inequality for the partial Fourier sum operator

I'm studying harmonic analysis by myself. One of the online notes gives the following claim as a remark: For any $N \in \mathbb{Z}^{+}$, let's use $S_{N}$ to denote the partial ($N$ terms) Fourier sum ...
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1answer
85 views

Analogous form of Hardy-Littlewood maximal inequality (weak/strong type) on affine subspaces

I'm using some online notes (Professor Schlag, Yale University) to study harmonic analysis by myself. He introduced the following claim as an exercise: For any function $f \in L^{1}(\mathbb{R}^{d})$ ...
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1answer
154 views

“Reversed” Bernstein Inequality

I'm studying harmonic analysis by myself, and I read some online notes that introduce the Bernstein inequality. One of them mention a reversed form of the Bernstein inequality, which is stated below: ...
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2answers
226 views

On inverting characteristic functions

Let $X$ be a random variable in $\mathbb{R}^n$ with distribution $\mu$ and characteristic function $\varphi$ (i.e. $\varphi(t)=\mathbb{E} e^{i\langle t,X\rangle}$). The standard inversion formula ...
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156 views

Haar mesure on $\mathrm{GL}_{d}(F)$

$\DeclareMathOperator\GL{GL}$Let $F$ be a $\mathfrak{p}$-adic field and $\mathscr{O}_{F}$ its valuation ring. For any measurable subset of $M_{d}(F)$ such as $$ A= \left( \begin{array}{ccc} a_{11}+t^{\...
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1answer
67 views

Show a inequality in homogeneous Besov space

How to prove $$ \lVert uv\rVert_{\dot{B}^{\frac{N}{p}-1}_{p,1}}\leqslant C \lVert u\rVert_{\dot{B}^{\frac{N}{p}}_{p,1}} \lVert v\rVert_{\dot{B}^{\frac{N}{p}-1}_{p,1}}$$ when $N\geqslant2 $and$1\...
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78 views

Span of parabolic inductions of discrete series representations

Let $G$ be the $\mathbf{Q}_p$-points in a $p$-adic reductive group, and let $R(G)$ be the Grothendieck group of the category $\mathrm{Rep}(G)$ of finite-length admissible smooth complex ...
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62 views

When can one bound the Hilbert transform on the torus in $L^1$?

If $f$ is a function on $\mathbb T$ then the conjugate $\tilde f$ of $f$ satisfies Zygmund's bound $$\int |\tilde f|\leq A \int |f|\log(e+|f|)+B.$$ I am curious if there are any conditions where one ...
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172 views

Finding (and saturating) a sharp Babenko-Beckner inequality for finite fields

My question is a follow-up to Abdelmalek Abdesselam's recent post What makes Gaussian distributions special? Local field version? asking about various characterizations of (real-valued) Gaussian ...
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53 views

About definition of NTA domain

I'm not an expert in analysis on very rough domains, such as NTA(Nontangentially Accessible Domain). Here is my question. Usually, NTA domain $\Omega$ is a domain that has inner and outer corkscrew ...
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1answer
250 views

Wiener Corollary in “An introduction to harmonic analysis” by Yitzhak Katznelson

I can't understand a lemma in "An introduction to harmonic analysis" by Yitzhak Katznelson which is stated as follows: Corollary. Let $\mu\in M(\mathbb T)$. Then $$\sum\limits_{\tau\in\...
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1answer
104 views

A question on a simple integral with a singular kernel?

I asked this question on math.stackexchange: Does this integral converge when $\frac{1}{p}+\frac{1}{q}\ge1$? No answers or very useful comments there. May be it is more appropraite for mathoverflow. ...
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1answer
142 views

Random Fourier series in Hilbert space

Let $H$ be a Hilbert space, and $X_n$, $ n\in \mathbb {Z}$, be a sequence of independent Bernoulli random variables $P(X_n = \pm 1) = \frac 12$. Is there a characterization of the sequences $a_n$, $ n\...
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1answer
219 views

Find strictly subharmonic function that vanishes at infinity

I am not sure about the term "strictly" subharmonic. What I want is a function $\psi\in C^{\infty}(\mathbb{R}^3)$ with $\Delta\psi>0$ and $\lim\limits_{|x|\rightarrow\infty}\psi(x)=0$. I ...
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2answers
666 views

Is there a nice orthogonal basis of spherical harmonics?

Recall that a function is harmonic if its Laplacian is zero. Let $\mathrm{Harm}(n,k)$ denote the vector space of $n$-variate harmonic polynomials that are homogeneous with degree $k$. When working ...
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86 views

Pushforward of measures with Fourier decay

Suppose $\gamma: [0,1]^d \to \mathbf{R}^{d+1}$ is a smooth map with nonvanishing Gaussian curvature, and $\mu$ is a probability measure compactly supported on $(0,1)^d$ such that $|\widehat{\mu}(\xi)| ...
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67 views

Dot product of functions on cosets

Some time ago I asked this same question at Math Stackexchange, because I thought that the question is nearly elementary. To my surprise, it was never answered. So I am elevating it to MathOverflow. I ...
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47 views

A question about super-harmonic functions

Lets call a function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ to be super-harmonic if $\nabla ^2 f = \sum_{i=1}^n \partial_i^2 f \leq 0$. Now given such a $f$ as above I want to consider the ...
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67 views

Reference requests: $W_{p}^1$-estimate for $(\triangle -\lambda)$ on Lipschitz domains

Let $1<p<\infty$ and $\lambda>0$. When $\Omega$ is a bounded $C^1$ or a bounded Lipschitz domain with small Lipschitz constant in $\mathbb{R}^d$, then for every $f\in L_p(\Omega)$ and $\...
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47 views

Does every compact abelian group contain a Kronecker set generating a dense subgroup?

Let $G$ be a compact metrizable abelian group with infinite exponent. Let $S^1 = \left\{z \in \mathbb{C} : |z| = 1 \right\}$. A set $K \subset G$ is a Kronecker set if, for every continuous function $...
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65 views

Smoothing property of a certain singular integral operator of non-convolution type

For simplicity, suppose that the dimension $d=2$, and let $g_s(x)$ be the Coulomb or Riesz potential defined by $$g_s(x) := \begin{cases} -\frac{1}{2\pi}\ln|x|, & {s=0} \\ c_s|x|^{-s},& {0<...
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1answer
71 views

Induced representations: space of continuous functions on $G$ to a Hilbert space

This question was asked in math stack exchange but received no replies: https://math.stackexchange.com/questions/4000655/induced-representations-space-of-continuous-functions-on-g-to-a-hilbert-space ...
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100 views

Integration by parts formula for the spectral fractional Laplacian

Let $f,g:C^\infty_c(0,1)$. Is there a formula similar to $$ \int_0^1 (f_{xx})^2g \ dx = \int_0^1 \frac{1}{2} (f_x)^2 g_{xx} \ dx - \int_0^1 f_{xxx}f_x g \ dx $$ for the spectral fractional Laplacian ...
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1answer
144 views

Using Fourier series to prove $-\int_0^1 u_{xxx}u_x \eta = \int_0^1 (u_{xx})^2\eta - \int_0^1 \frac{1}{2} (u_x)^2 \eta_{xx}$

Let $u, \eta$ be smooth functions and $\eta$ compactly supported in $(0,1)$. Integrating by parts, we can easily prove $$-\int_0^1 u_{xxx}u_x \eta = \int_0^1 (u_{xx})^2\eta - \int_0^1 \frac{1}{2} (u_x)...
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146 views

Functions whose Fourier coefficients satisfy $ \sum_{k=1}^\infty |c_k| < 1 $?

Let $f:(0,1) \to \mathbb R$ be a function that can be written as $$f(x) = \sum_{k=1}^\infty c_k \phi_k(x),$$ where $\phi_k(x) = \cos(\pi k x)$. What is the minimal assumption required on $f$ to ...
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60 views

Non-Hausdorff part of measure zero?

Let $G$ be a locally compact group of type I. By Theorem 4.4.5 of Dixmier's book on C*-algebras, there exists a dense open subset $U$ of the unitary dual $\widehat G$, which is a Hausdorff space. Can ...
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1answer
119 views

Mean value principle reversed

Suppose that $\Omega \subset \mathbb R^3$ is a domain with smooth boundary $\partial \Omega$ and suppose that $0\in \Omega$. Given any $f \in C^{\infty}(\partial \Omega)$ let $u^f$ denote the unique ...
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1answer
86 views

Twisted winding number

Consider the contour integral $\frac{1}{2\pi i}\oint_\gamma\chi(z)\frac{dz}{z}\,,$ where $\gamma$ is a (not necessarily simple) closed curve lying in $\mathbb{C}\setminus{0}$ and $\chi\colon\mathbb{...
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2answers
277 views

Sum of $\sin$ when angles shrink by $1/n$

There are many identities known like $$\sum_{k=0}^{n-1} \sin (k \cdot \theta + \varphi) = \frac{\sin\left(n \cdot \frac{\theta}{2}\right)}{\sin\left(\frac{\theta}{2}\right)} \cdot \sin \left(\frac{2 \...
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155 views

Arbitrarily long compositions of a functions that stay within $[0,1]$

This question is inspired by this MSE post. (in short, it is a continuous strengthening of the case of their claim for $k=1$, which is the only case which has been fully resolved discretely) Given $c\...
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125 views

Harmonic analysis on reductive groups over $\mathbb{R}$

A common way of doing harmonic analysis on (the $\mathbb{R}$-points of) reductive groups over $\mathbb{R}$ seems to be to use results from semisimple groups and "see what happens on the center&...
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222 views

The Cauchy Transform, and the convergence of the Fourier-Stieltjes transforms of a sequence of measures

Let $C\left(\mathbb{R}/\mathbb{Z}\right)$ denote the Banach space of continuous, $1$-periodic complex-valued functions on the unit interval, let $M\left(\mathbb{R}/\mathbb{Z}\right)$ denote its dual, ...
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1answer
47 views

Integrability of $\exp\left(p\int_0^t |w(s,x(s,y))| \mathrm{d}s\right)$ for $w\in L^\infty(0,T;BMO(\mathbb{T}^d))$

Let $w\colon [0,T]\times\mathbb{T}^d \to \mathbb{R}^n$ be such that $$ \|w\|_{L^\infty(BMO)} := \sup_{t\in[0,T]}\|w(t,\cdot)\|_{BMO} \leq C $$ and $\int_{\mathbb{T}^d} w(t,x)\mathrm{d}x = 0 $ for all $...
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105 views

Predual of $BMO(\mathbb{T}^d) $

In 1971, Fefferman characterized the predual of $BMO(\mathbb{R}^d)$ as the Hardy space $H^1(\mathbb{R}^d)$. Is there a characterization of the predual of $BMO(\mathbb{T}^d$)?
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1answer
96 views

A Frostman-type result for measures satisfying uniform lower density conditions

Let $\mu$ be a finite, compactly supported, non-zero measure on $\mathbb{R}^d$ for an integer $d$. Let $B(x,r)$ denote the ball of radius $r>0$ centered at $x \in \mathbb{R}^d$. For $\delta \in [0,...
5
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1answer
171 views

A simple oscillatory integral with a non-smooth phase

Let $\phi\in C_c^\infty(\mathbb{R})$ be an even function such that $\chi_{(-1/2,1/2)}\le\phi\le \chi_{(-1,1)}$, where $\chi_{(a,b)}$ stands for the indicator function of the interval $(a,b)$. For $\...
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1answer
162 views

Beltrami equation with harmonic coefficient

I need to find solutions to the Beltrami equation $$ \frac{\partial w}{\partial\overline{{z}}}=e^{i\varphi(z)}\frac{\partial w}{\partial z} $$ for $w=w(z,\overline{z})$ and $\varphi(z)$ some given, ...
3
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0answers
72 views

Uncertainty principle, Sobolev embedding, and norm estimates

Terence Tao offers a nice discussion of different function spaces in this blog. In the blog there is an explanation of the tradeoff between regularity $s$ and integrability $p$, where $s,p$ are ...
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2answers
276 views

Existence of $f \in L^2(\Bbb R^n)$ with $f=g_1$ on $E$ and $\mathscr{F}(f)=g_2$ on $F$

The question has been posted here but had no response. Question: Suppose $E,F$ subsets of $\Bbb R^n$ have finite measure. Show that for any $g_1,g_2 \in L^2(\Bbb R^n)$ there exists $f \in L^2(\Bbb R^...

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