Questions tagged [harmonic-analysis]

Harmonic analysis is the generalisation of Fourier analysis. Use this tag for analysis on locally compact groups (e.g. Pontryagin duality), eigenvalues of the Laplacian on compact manifolds or graphs, and the abstract study of Fourier transform on Euclidean spaces (singular integrals, Littlewood-Paley theory, etc). Use the (wavelets) tag for questions on wavelets, and the (fourier-analysis) for more elementary topics in Fourier theory.

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18 views

A question about Fourier multipliers

The question The question is 2.5.10 from Grafakos' Classical Fourier Analysis. It starts by defining a multiplier $$m_{\text{per}}(\omega)=\sum_{k\in\mathbb Z^d}m(\omega -k),$$ where $m$ is a Fourier ...
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15 views

Poisson summation formula as a special case of the trace formula

For $f \in L^1(\mathbb R)$, the Fourier transform $\hat{f}: \mathbb R \rightarrow \mathbb C$ is defined by $$\hat{f}(x) = \int_{\mathbb R} f(y) e^{2\pi i xy}dy.$$ The Poisson summation formula asserts ...
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A weighted estimate for Hardy-Littlewood maximal functions

The set up \begin{align*} &1<p\leq q<\infty,\\ &f\in L^{p}(0,\infty),\;f\geq 0,\\ &Mf(x):=\sup_{r>0}\frac{1}{|B(x,r)|}\int_{B(x,r)} f(y)dy. \end{align*} Here $B(x,r)$ is the ...
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Question about Proof of Wold's Decomposition Theorem (Operator Theory ) on Wikipedia

(Wold-decomposition theorem 1954.) Every isometry is a direct sum of unitary and unilateral shifts. Sketch of Proof: https://en.wikipedia.org/wiki/Wold%27s_decomposition#A_sequence_of_isometries In ...
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Super-level sets of Hardy–Littlewood maximal function are open?

I am working on the book Measure, Integration and Real Analysis by Sheldon Axler. I am stuck on Problem 9 of Section 4A. For $h: \mathbb{R} \to \mathbb{R}$ Lebesgue measurable, $h^*$ is defined as ...
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37 views

Find the eventually periodic points of the function $f(x)=\lvert x-1 \rvert$

Let $\emptyset\neq X\supset A$, $f:A\rightarrow X$, a point $x\in A$ is eventually periodic point of $f$, if : $$\exists n_0\in\mathbb N \exists N\in\mathbb N: f^{n+N}(x)=f^n(x) \quad \forall n\ge ...
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+150

Stokes theorem for a current

I am struggling to understand the Stokes theorem for currents(differential form with distributional coefficients). The statement is as follows: Lest $S$ be a current of degree $N - 1$ with compact ...
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Restriction-Extension Duality when $q = 1$

The restriction conjecture says that for $1 \leq p < 2d/(d+1)$ and $(d+1)(1/p) - 2 \geq (d-1)(1 - 1/q)$, and any $f \in C_c(\mathbf{R}^d)$, $$ \| \widehat{f} \|_{L^q(S)} \lesssim \| f \|_{L^p(\...
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Do Hodge $*$-operators glue?

For ${\Bbb P}_{\Bbb C}^2 = \underset{i = 0, 1, 2}{\bigcup} {\Bbb A}_{i, \Bbb C}^2$, we have Hodge $*$-operators on each affine open ${\Bbb A}_{i,{\Bbb C}}^2$. For example $z_1 = X_0/X_2, z_2 = X_1/X_2$...
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Properties of good kernels (Convolution of good kernel with function continous at a point)

Let $(K_t)_{t>0}$ be a family of nonnegative functions such that $$(i) \int_{\mathbb{R}^n}K_t(x)\text{d}{y}=1 \;\;\;\text{for all} \;\;\;t>0$$ $$(ii) \;\text{For every} \;\epsilon>0 \;\;\;\...
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Wiener Corollary in “An introduction to harmonic analysis” by Yitzhak Katznelson

I can't understand a lemma in "An introduction to harmonic analysis" which is stated as follows: Corollary. Let $\mu\in M(\mathbb T)$. Then $$\sum\limits_{\tau\in\mathbb T}|\mu(\{\tau\})|^2=...
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Are there any real-world applications for the logistic map (as defined in chaos theory)?

Is the logistic map only useful for theoretically exploring nonlinear dynamical systems, or can it be applied to real-world scenarios in any practical way?
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Order of magnitude in Absolutely convergent Fourier series

There is a question in "An introduction to harmonic analysis" by Yitzhak Katznelson. Problem: Let $f\in A(\mathbb T)$($\mathbb T$ denotes the torus), i.e. $\sum\limits_{n\in\mathbb Z}|\hat f(...
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When does the conjugation action preserve Haar measure?

Let $H\trianglelefteq G$ be a normal closed inclusion of locally compact groups. Then $G$ acts by conjugation on $H$, thus on the one-dimensional space of Haar measures on $H$. For this action to be ...
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Harmonic analysis and characters

In the following video, at the linked time, https://youtu.be/HhTGyDuNI_w?t=762 there is an argument presented for characters and the discrete fourier transform. Is there some book where I can find ...
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34 views

Haar analysis decay rate

Question Prove the partial Haar sum $P_M(f_N)$, defined by $P_M(f_N)=\sum_{0\leq j < M} Q_j(f_N)+P_0(f_N)$, decays exponentially in $M: ||f_N-P_M(f_N)||_{L^2([0,1))} = 2^{-M/2}$, where $f_N=2^N\...
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Fourier transform following an decay rate of $1/\sqrt{M}$

The Question Show that if we view $f_N = 2^N\chi_{[0,2^{-N}]}$ as a periodic function on $[0,1),$ with $M^{th}$ partial Fourier sum $S_M(f_N)(x)=\sum_{|m|\leq M}\hat{f_N}(m)e^{2\pi imx},$ then the ...
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25 views

Find strictly subharmonic function vanishing 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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Do $L^p$ integrals over fixed radius balls satisfy this reverse Hölder-like inequality?

For measurable $f:\mathbb R^n \to \mathbb R$, define$$M_qf(x) =\left(\int_{B_{1/2}(0)} |f(x+y)|^{q} dy \right)^{1/q} = \|f\|_{L^q(B_{1/2}(x))} $$ and define the (inhomogeneous) dyadic annuli $ D_0 := ...
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Show that ${(1,0,0),(2,1,0),(3,2,1),(4,3,2)}$ is a frame for $H=R^3$.

Show that ${(1,0,0),(2,1,0),(3,2,1),(4,3,2)}$ is a frame for $H=R^3$. I'm confused on how to show it, please help me Definition
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Operator which commutes with every translation also commutes with convolutions

I am currently trying to prove that every bounded $T:L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)$ which commutes with translations must be a Fourier multiplier. I've seen a few posts that deal with this ...
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Problems about Wiener algebra(absolutely convergent fourier series)

The Wiener algebra $A(\mathbb T)$ is defined as the space of all functions defined on $\mathbb T$(the torus) such that its fourier series satisfies: $$\|f_n\|_{A(\mathbb T)}:=\sum\limits_{n}|\hat f(n)|...
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Reference request: Commutator estimate

I am wondering if the following commutator estimate is true, and in such case, where can I find a proof for it. Let $N\in 2^{\mathbb{Z}}$ dyadic, and let's denote by $P_N$ the standard Littlewood-...
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1answer
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Inner product of two matrix coefficient functions of an irreducible representation.

$\newcommand{\ab}[1]{\langle #1\rangle}$ $\newcommand{\C}{\mathbf C}$ Let $G$ be a compact Hausdorff group and $\mu$ be the normalized Haar measure on $G$. Let $\sigma:G\to \mathscr U(V)$ be a finite ...
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34 views

Prove $\{z^n|n \in \mathbb{Z}\}$ is an orthonormal basis of $L^2(S^1)$ with Haar measure

Prove $\{z^n|n \in \mathbb{Z}\}$ is an orthonormal basis. We show that this is an orthonormal set. All we need to show is that $\int_{S^1} z^n d\mu=0$ since $\mu$ is a Haar measure $\int_{S^1} z^n d\...
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27 views

When does this $l^s$ inequality hold in Lorentz Spaces

For which parameters $0 < p,q,s \leq \infty$ do we have a bound $$ \left\| \left( \sum_{n = 1}^\infty |f_n|^s \right)^{1/s} \right\|_{L^{p,q}(X)} \lesssim_{p,q,s} \left( \sum_{n=1}^\infty \| f_n \|...
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118 views

How complicated can Lipschitz domains be?

A Lipschitz domain $\Omega$ is a domain in $\mathbb R^n$ whose boundary $\partial\Omega$ is locally the graph of a Lipschitz continuous function. For example, any $C^1$ domain is Lipschitz and a cube ...
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35 views

Is this a classical result of harmonic analysis?

I am reading a paper and I really don't understand the following how hard I think about it. For a $\nu \in \mathbb{R}_+$, let $v_\nu$ be the solution of the following incompressible Navier-Stokes ...
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35 views

Conditions that satisfy Fourier in a particular situation

Proposition: Complex sequence ${a_n\;}$,${b_n\;}$($n$ from $0$ to $∞$),For a number $C>0$, $|a_0\;|≤C$, $|a_n\;|≤Cn^{-k}$,$(n ≥ 1)$,$|b_n\;|≤Cn^{-k}$,$(n ≥ 1)$ If it meets, the series$$\frac{a_0\;}{...
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1answer
67 views

Compact Fourier multiplier?

Let us take some continuous $m\in L^\infty(\mathbb R^d)$ and suppose $m(\xi)\to 0$ as $|\xi|\to \infty$. Let $T_m:L^2(\mathbb R^d)\to L^2(\mathbb R^d)$ be the associated Fourier multiplier operator, i....
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Dot product of functions on cosets

Let the measures of locally compact groups $\,K < G\,$ be $\, dk\,$ and $\, dg\,$, correspondingly. For a Hilbert space $\mathbb{V}$ equipped with a dot product $\,\langle~,~\rangle\,$, introduce ...
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How to find eigenfuctions of shifts operator and dilation operators

I am trying to find eigenfunction of shift operators and dilation operators. Shift operators $S_t$ maps function $f(x)$ to $f(x+t)$. I am trying to find continuous function $f$ such that $(S_tf)(x) = \...
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Estimation of Herglotz Function

A Herglotz function $F(z)$ is a holomorphic function form $\mathbb{C}_+$ to $\mathbb{C}_+$,where $ \mathbb{C}_+=\left\{ z:\text{Im}z>0 \right\} $.We have known that $F(z)$ has a representation $$F\...
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Calculate the second harmonic contribution from a wave-like signal

I have a wave signal ($M_z$) computed from a simulation which has a main frequency of $\nu\approx0.158[GHz]$, and I want to extract the second harmonic contribution $M_z^{2\omega}$ of the signal. I ...
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1answer
80 views

Unique continuation at the boundary for harmonic functions in the plane

Consider the set $U = (-1,1) \times \{ 0\} \subset \mathbb R^2$ and a continuous function $f : U \rightarrow \mathbb R$. Then for any domain $\Omega \subset \mathbb R^2$ such that $U \subset \partial\...
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1answer
35 views

Fourier Transform of Even/Odd Complex Functions

For any complex-valued $f \in L^1(\mathbb{R})$, let's define its Fourier transform $\hat{f}$ with the following convention $$ \hat{f}(\omega) := \int_{\mathbb{R}} f(x) e^{-i \omega x} dx $$ I would ...
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1answer
101 views

Compute the difference of three series (motivated by an integral formula)

Let \begin{align} L&=2^{3 / 2} \pi^{4} \frac{1}{4} \sum_{n, m, k=1}^{\infty} n^{2} m a_{n} a_{m} b_{k}\left(\delta_{m, n+k}+\delta_{n, m+k}-\delta_{k, n+m}\right) \\[5pt] R_{1}&=2^{3 / 2} \pi^{...
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A_2 muckenhoupt weights

My question is easy. Does A_2 muckenhoupt condition hold even if the weight $\omega$ is zero on some sets? What is the condition in this case? I did not find a clear reference or explanation about ...
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41 views

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

EDIT: Answered in MathOverflow @https://mathoverflow.net/questions/382324/induced-representations-space-of-continuous-functions-on-g-to-a-hilbert-space Let $G$ be a locally compact group, $H$ a ...
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BMO/BMOA spaces

I am working with $BMO$ and $BMOA$ spaces (Let us assume in the disk). I am asking myself if it is true that $BMO=BMOA + \bar{BMOA}$, where with $\bar{BMOA}$ , I mean functions in $BMOA(\mathbb{C}\...
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Discrete Fourier transform in Distribution sense

When we describe the Fourier transform, we first start with nice functions $S(R^n)$ and then extend the operator continuously to a bigger space $S'(R^n)$, which is the space called tempered ...
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An approximation of a d-dimensional function using an order q<d function

Suppose we have a function $f:\mathbb{R}^d\to\mathbb{R}$. Using the notation of https://www.maths.unsw.edu.au/sites/default/files/amr08_5_0.pdf, which defines an order $q$ function $g:\mathbb{R}^d\to\...
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1answer
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let $\sum_{k=0}^{n}|f_{k}|^{2}=g$ then if g is constant then $\forall k\in[n].f_{k}$ is constant [duplicate]

let $\sum_{k=0}^{n}|f_{k}|^{2}=g$ and then if g is constant then if $\left\{ f_{k}\right\} _{k=1}^{n}$ is holomorphic then : $\forall k\in[n].f_{k}$ is constant. My try: Each of the $f_k$ is has ...
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54 views

$C^*$ algebra of LCA-group

I'm reading A. Deitmar and S. Echterhoff's book "Principles of Harmonic Analysis", and have a confusion of the proof in the snapshot: Why do we need to prove $\|f\|\equiv\|f\eta\|$ in order ...
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1answer
42 views

Topological groups: if $A$ is an open subset of $G$, then $A^{-1}$ is open.

Let $G$ be a topological group with open subset $A$. I want to prove very thoroughly/precisely that $A^{-1}:=\{a^{-1}:a\in A\}$ is also open. Obviously the key to this proof is the fact that the ...
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1answer
40 views

a series of functions that converges uniformly, converges to a harmonic function and so its derivative

let $u_{n}:G\rightarrow\mathbb{R}$ be a series of harmonic on a domain G, and $u_{n}\rightarrow u$ uniformly. Then u must be also harmonic. it was written here, but not formalize:Does a uniformly ...
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20 views

Littlewood-Paley identity

Denote $\mathbb{D}$ the unitary disk on $\mathbb{R}^2$ and $\mathbb{T}$ its border. Suppose $u$ is a $C^2$ function in $\mathbb{D}$ and continuous in $\bar{\mathbb{D}}$ with $u(0)=0$. If I understood ...
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1answer
29 views

Harmonic function defined through harmonic functions

Let $u(x,y)$ and $v(x,y)$ be a pair of conjugate harmonic functions, and let $x(u, v)$ be a harmonic function of the variables $u$ and $v$. Prove that $\chi [u(x,y),v(x,y)]$ is a harmonic function of ...
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32 views

Why do atoms belong to the real Hardy space $\mathcal{H}^{1}(\mathbb{R}^n).$

Define atom $a\in L^1(\mathbb{R}^n)$ associated to the ball $B_r(x_0)$ such that it satisfies $\text{supp}(a)\subset B_r(x_0).$ $|a|\leq 1/|B_r(x_0)|$ and so $||a||_{L^1}\leq 1.$ $\int_{B_r(x_0)} a ...
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32 views

Potential theory and examples of hyperharmonic functions

This follows the book "Stratified Lie Groups and Potential Theory for Their Sub-Laplacians" by Bonfiglioli, Lanconelli, and Uguzzoni. In chapter 6, the book introduces abstract harmonic ...

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