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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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The scaling effect on the Besov norm

The Besov spaces can be defined using the dyadic decomposition. For this purpose, let $\varphi \in C^\infty_c(\mathbb R^n)$ be such $ {\rm supp}(\varphi) \subseteq \left\{x; \, \frac 12< |x| &...
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The $L^\infty$ boundedness of the resolvent of Harmonic Oscillator in terms of seminorms

I am reading Watanabe's book "Lectures on Stochastic Differential Equations and Malliavin Calculus". In Page 48, it is said that by means of (damped) harmonic oscillator $1+|x|^2-\Delta$, we ...
ze min jiang's user avatar
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Integral of complex exponential with phase given by multiplication by a rational

I want to check that if $f \in L^1(\mathbb{T})$, $m$ is a positive integer, and $f_m(t) = f(mt)$ then $$\widehat{f_m}(n) = \begin{cases}\hat{f}(n/m) &\text{ if } m \mid n, \\ 0 & \text{ if } m ...
approximate-identity's user avatar
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Linking Fourier Coefficients of periodic functions

Let $\tau\in (0,1)$ and assume that we have a $\tau$-periodic function $$f_1(t) = \sum\limits_{k\in\mathbb{Z}} a^1_k e^{\frac{2\pi i k}{\tau}t},$$ a $(1-\tau)$-periodic function $$f_2(t) = \sum\...
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Estimation of Maximal Functions

Assume that $f\in L_{\mathrm{loc}}^1\left(\mathbb{R}^n\right).$ Proof:if$f(x)=O(|x|^{-n})\left(|x|\right.\to\infty)$, then $Mf(x)=$ $$O(|x|^{-n}\log|x|)\:(|x|\to\infty).$$ I think we can use this ...
moo poo's user avatar
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The conjugate of a cosine series is a sine series

In Katznelson's An Introduction to Harmonic Analysis the author defines the conjugate $\widetilde{S}$ of a trigonometric series $$S \sim \sum_{n = \infty}^\infty a_n e^{inx}$$ by $$\widetilde{S} \sim \...
approximate-identity's user avatar
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Decay estimate for a heat-like kernel.

Consider a integral kernel $G(x,t)$, which satisfies $$\hat G(\xi ,t)\le Ce^{-c|\xi|^2t},$$ for some positive constant $c,C>0$. I want to show that it satisfies the following decay estimate for ...
Varnothing S's user avatar
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The asymptotic of an integral $I$

Consider the integral $$ I(\lambda)=\int_0^1 \frac{1}{\sqrt{v}}\,\left( \int_{-\infty}^{+\infty} \frac{e^{i\lambda u (u^2-v)}}{\sqrt{u^2+ 4v}}\,\varphi(u,v)\, du\right) dv, $$ where $\varphi\in C_0^\...
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Maximal function on mixed $L^{p}$ spaces

Consider $ f_{j,k}$ to be a function in $L^{p}(l^{q}(l^{2}))$, that is $$ \Vert f_{j,k} \Vert^{p}_{L^{p}(l^{q}(l^{2}))} = \int_{\mathbb{R}^{n}} \left( \sum_{k} \big[ \sum_{j} \vert f_{j,k}(x) \vert^{2}...
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Complex exponential Fourier coefficients of a convolution involving the exponetial function

In the book "Elementary Classical Analysis", by Marsden, the following is proposed as a worked example: Let $f:[0,2\pi]\to\mathbb{R},g:[0,2\pi]\to\mathbb{R}$ and extend by periodicity. ...
Pablo Álvarez's user avatar
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extending weak estimate

Let $T: L^{1}\left(\mathbb{R}^{n}\right) \cap {L^2}\left(\mathbb{R}^{n}\right) \rightarrow L^{1}\left(\mathbb{R}^{n}\right)$ be a linear operator which is of weak type $(1,1)$. Show that $T$ extends ...
Document123's user avatar
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Are the famous operators S_n of Fourier series theory continuous?

In a previous Q&A of mine (this here) I mentioned what Katznelson calls homogeneous Banach spaces on the torus $\mathbb T = \mathbb R / 2\pi \mathbb Z $ (I mention him here because others define ...
Ulysse Keller's user avatar
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Does the maximal function of a positive function say away from zero on bounded sets?

I was reading the book "Fourier Analysis" by J. Duoandikoetxea and D. Cruz-Uribe. In one of the results, they want to prove the following weighted inequality: Theorem (Weighted Inequality): ...
Aniruddha Deshmukh's user avatar
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Singular integral, Navier-Stokes equations.

Given $f \in L^2((0,T))$, I would like to prove that $$ \int_0^t (t-s)^{-\frac{1}{2}}|f(s)|ds $$ is in $L^2((0,T))$ as well. Is it true? What seems difficult to me is that it does not look like as ...
Sergio Scalabrino's user avatar
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If $\phi \in L^2(\mathbb T)$ and $\widehat{\phi}(\mathbb Z_{<0}) = \{0\}$, then $\phi \ne 0$ a.e. or $\phi = 0$ a.e. on $\mathbb T$

I would like to show the following result. Suppose $\phi\in L^2(\mathbb T)$ satisfies $\widehat{\phi}(n) = 0$ for all $n\in \mathbb Z_{<0}$. Then, either $\phi \ne 0$ a.e., or $\phi = 0$ a.e. The ...
stoic-santiago's user avatar
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Asymptotic estimate of the inverse Fourier transform of $e^{-\xi^{2q}}$

In V. Yu. Krylov's paper [1], he estimates the inverse Fourier transform of $e^{-\xi^{2q}}$ using harmonic measure arguments (which I don't understand since the reference is written in Russian ...
Mango Warrior's user avatar
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Has every homogeneous Banach space on the torus an operator S_n (Katznelson on Harmonic Analysis)?

In his book about Harmonic Analysis Katznelson defines homogeneous Banach spaces on the 1-dim. torus: in short, such a space is a linear subspace $B$ of $L^1(\mathbb T)$ with a norm denoted $||f||_B$ (...
Ulysse Keller's user avatar
5 votes
2 answers
113 views

Normal character on a group von Neumann algebra

For a locally compact group $G$, I will denote by $L(G)$ its group von Neumann algebra, which is the von Neumann algebra acting on $L^2(G)$, generated by the image of left regular representation $\...
Mogget's user avatar
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Equality between rings of invariant differential operators

Let $G$ be a real connected Lie group in the Harish-Chandra class [GV - Definition 2.1.1]. Let $\mathfrak{g}$ denote its Lie algebra. This is a reductive Lie algebra and we have $\mathfrak{g} = \...
Sentem's user avatar
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Hörmander's proof of Beurling's uncertainty principle and the application of Phragmén-Lindelöf

I am trying to understand the proof in the following paper. https://projecteuclid.org/journalArticle/Download?urlId=10.1007%2FBF02384339 where it is proven that if $$\int \int_{R^2}|f(x)\hat{f}(y)|e^{|...
Valsinator's user avatar
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Can we prove Young's Convolution Inequality only using Interpolation?

I tried to prove Young's Convolution Inequality $\|f\ast g\|_r\leq \|f\|_p \|g\|_q$ for $1/r+1=1/p+1/q$, where $p,q,r\in[1,\infty]$. From the Riesz–Thorin Interpolation theorem, it suffices to prove ...
Confusion's user avatar
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confusion on understanding how calderon zygmund decomposition

I am not able to see how by Calderon zygmund decomposition. We have $|\{x \in \mathbb{R}^{n}: M_{D}g(x) > \lambda \}| \leq \frac{1}{\lambda} \int_{\{M_{D}g(x) > \lambda\}}|g(x)|dx$. Here $M_{D}$ ...
Document123's user avatar
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Constant in the Banach valued Marcinkiewicz-Zygmund inequality

Let $(\xi_i)_{i=1}^n$ be a sequence of real-valued independent random variables with zero mean. The classical Marcinkiewicz-Zygmund inequality is the following \begin{align} (E(\sum_{i=1}^n\xi_i)^p)^{...
alphabeta's user avatar
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Derivatives of Poisson kernel.

In Stein book Singular integrals and differentiability properties, page 86 the following is written: Let $f \in L^{p}$ and $ u(x,y) = P_{y} * f (x) $ Where $P_{y}$ is the Possion kernel. then it is ...
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disk of convergence of composition of p-adic functions

I watched a video that sketched out the regions of convergence for both the p-adic logarithm and the p-adic exponential functions. I thought about all this and asked myself: How would I find the ...
John Zimmerman's user avatar
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Regularity conditions for properties of Hilbert transform

I want to use a property of Hilbert transform proved in the following paper, see Lemma 3.1 in https://arxiv.org/pdf/2009.01882.pdf. My question is what does "sufficiently regular" mean in ...
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Showing that any harmonic tempered distribution is a polynomial.

In PDE basic theory by the author Taylor I can't understand two facts: Prop 1. If $u\in \mathcal{S}'(\mathbb{R}^n)$ is supported by $\left\{0\right\}$, then there exists $k$ and complex numbers $a_\...
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Is this summation over Fourier coefficients a meaningful way to compute dot product despite its weaknesses? Or what is the better way to define it?

So let's say we have two vectors $a \in \mathbb R^n$ and $b \in \mathbb R^m$. Let $L=\text{lcm}(n,m)$. Consider the natural inclusions $a,b \mapsto \mathbb R^L$, which simply continually concatenate $...
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Vanishing of an exponential sum $f(x)=\sum_{k=1}^n a_k e^{ik b_k}$ on sets of the form $\{ \alpha l : l = -m, \dots, m \}$

Suppose that $f(x)=\sum_{k=1}^n a_k e^{ik b_k}$ is an exponential sum with frequencies $b_k \in \mathbb R$ and coefficients $a_k \in \mathbb C$. Further, let $S(\alpha, m)$ be the set $$ S(\alpha, m) =...
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More Efficient Version of Vitali Covering Lemma

This question is a response to the answer given in this SE post, concerning the use of the Vitali covering lemma to prove the Hardy-Littlewood maximal inequality in general dimension. For reference, ...
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Does this proof for product estimate in Sobolev spaces work?

For $u,v\in \mathcal{S}(\mathbb R^d)$, $s,s_1,s_2\in\mathbb R$, $$ \|uv\|_{H^s}\lesssim_{d,s,s _1}\|u\|_{H^{s_1}}\|v\|_{H^{s_2}}. $$ where $s_1+s_2=s+d/2>0$. My attempts: We work in a direct ...
P0lyno3ial's user avatar
2 votes
1 answer
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Finding the harmonic conjugate of the function

$u(x,y)=e^{x^2-y^2}(e^y \cos(x-2xy)+ e^{-y} \cos(x+2xy))$ Solving the Cauchy-Riemann equations for this is not practical. Alternatively, I could try to express $u(x,y)$ in the form of the real or ...
Derewsnanu's user avatar
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A sequentially continuous linear operator on tempered distributions is continuous

The following proposition is true. My question is whether we can find a direct, pedagogical proof that can be delivered in class without having to introduce new concepts from functional analysis. In ...
Giuseppe Negro's user avatar
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Condition of function $f$ for non-only-positivity of its Fourier transform

I am looking for conditions on a function $f(x)$ for its Fourier transform $f(k)$ to not be always $\geq 0$ for all $k$; i.e. a condition on $f(x)$ such that $f(k)<0$ for some k, where $f(k)$ is ...
Alessandro Salvatore's user avatar
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What is the inclusion property between the Besov space $B^n_{\infty,1}(\mathbb{T})$ and $C^n(\mathbb T)$?

I am trying to understand the Besove space $B^n_{\infty, 1}(\mathbb T)$ on the unit circle. I also try to find some simple set inclusion of these subspaces with some known classical function space. I ...
CCCC's user avatar
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1 answer
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How to calculate $\sum_{n=1}^{\infty} (\int_0^\pi x^3cos(nx)dx)^2$

For each $n \geq 1$, denote $C_n= \int_0^\pi x^3cos(nx)dx$. Calculate $\sum_{n=1}^{\infty} {C_n}^2$ My suggestion: The given sequence is proportional to Fourier coefficients of $x^3$ expansion to an ...
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sum of rapidly decreasing functions over $S$-integers minus origin.

Let $F$ be a number field and $S$ a finite set of places including all infinity places. Let $f\in \mathcal{A}(F_\infty)$ be a rapidly decreasing function (in the usual sense), and $K\subseteq F_{S\...
hhh007's user avatar
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Definition of the Besov space

The Besov spaces can be defined using the dyadic decomposition. For this purpose, let $\varphi \in C^\infty_c(\mathbb R^n)$ be such $ {\rm supp}(\varphi) \subseteq \left\{x; \, \frac 12< |x| &...
A. PI's user avatar
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Sum of squares of sines (Normal modes of a harmonic chain) of different periods

I came across this summation while solving harmonic chain. \begin{align} \sum_{l=1}^N \sin^2{(\frac{m\pi}{N+1}\cdot l)} \end{align} for an integer m that can range from 1 to N. So the question is ...
betha manjunath's user avatar
1 vote
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Differentiable irreducible representations of the abelian group $(\mathbb{R}^d,+)$

In Woit's book chapter 20, he considers character of the group $(\mathbb{R}^d,+)$. Now he claims that irreducible representations of $(\mathbb{R}^d,+)$ are one dimensional and given by $\alpha_{\...
Donky Dang's user avatar
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1 answer
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An Estimate in Calderon Zygmund for Periodic Function

I am reading the following paper of Calderon and Zygmund http://matwbn.icm.edu.pl/ksiazki/sm/sm14/sm14122.pdf On page 3 (252) they provide an estimate (2.3) which is following $$ |K(x-x_{\nu}) - K(-...
pde's user avatar
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1 answer
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Harmonic conjugated functions and simply connected domain in wikipedia

In wikipedia (link), it says: (i) Therefore, a harmonic function $u$ admits a conjugated harmonic function if and only if the holomorphic function $g(z)\colon=u_{x}(x,y)-iu_{y}(x,y)$ has a primitive $...
studyhard's user avatar
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No Riemann-integrable function has the harmonic series has Fourier coefficients

This is a question that was [already asked on this site][1] but got no satisfactory answer. I would like to rephrase and show my own attempt. So the point is, consider the sequence $a_k = \begin{cases}...
confusedTurtle's user avatar
1 vote
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26 views

Explicite formula for Almansi decomposition

In 1899 E. Almansi proved that any polyharmonic function of order $p$, i.e. in the kernel of the differential operator $\Delta^p$, defined on a shar shaped domain $\Omega\subset\mathbb{R}^n$, can be ...
Giulio Binosi's user avatar
2 votes
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36 views

Reverse estimate of Hardy-Littlewood maximal function in Sobolev spaces

Given $f\in L^1_{loc}(\mathbb{R}^n)$, we consider the maximal function $$ Mf(x) := \sup_{r>0} \frac{1}{|B(x,r)|} \int_{B(x,r)} |f(y)|dy. $$ It is known that $M : W^{1,p}(\mathbb{R}^n)\to W^{1,p}(\...
Dorian's user avatar
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Formal Demonstration of an Integral Inequality Involving Nested Sets

I am delving into an analysis involving a specific integral inequality that arises within the context of Fourier analysis, particularly focusing on expressions involving a series of integrations over ...
Sara Testori's user avatar
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32 views

Absolutely continuous measures perserve under a continuous function

Let $f: X \to \mathbb{R}$ be a continuous function over a compact metric space $X$. Assume that $\mu$ and $\nu$ are two Borel probability measures on $X$. Suppose that $\mu << \nu$. Is it true ...
Adam's user avatar
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Periodic Hilbert Transform for integrable functions is well-defined (Katznelson)

I'm struggling to understand a proof in Katznelson (An Introduction to Harmonic Analysis, III - Lemma 1.3) Basically, given $f \in L^1(T)$, take $F(z) = e^{-f - i \tilde{f}}$. Since this is ...
user895052's user avatar
4 votes
1 answer
117 views

Method to produce polyharmonic functions from harmonic ones

Let $F:\mathbb{R}^2\to\mathbb{R}$ be a harmonic function (it can be thought as one component of a holomorphic function), take $m$ odd and let $f:\mathbb{R}^{m+1}\to\mathbb{R}$, $$f(a,x_1,\dots,x_m)=\...
Giulio Binosi's user avatar
2 votes
1 answer
28 views

Estimating Difference of Two Averages of Function by Its BMO Norm

Assume $f\in \text{BMO}(\mathbb{R}^d)$. Denote $f_Q=\displaystyle\dfrac{1}{|Q|}\int_Q f(x)\,dx$. I want to show for $\alpha>2$ and any cube $Q$ with positive volume, we have $|f_{\alpha Q}-f_Q|\leq ...
Laurence PW's user avatar

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