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, ...

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60 views

Is a bounded function always the Hilbert transform of some other function?

Given $f \in L^\infty(\mathbb R)$, does there always exist a $g$ (in some space) such that \begin{equation*} Hg=f, \end{equation*} where $Hg$ is the Hilbert transform of $g$ ? In other words, is the ...
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40 views

Unbounded Hilbert Transform of $C_{c}(\mathbb{R})$ Function

I want to give an example of a continuous, compactly supported function (denoted $C_{c}(\mathbb{R})$) $f$, such that the distribution $H(f)$, where $H$ is the Hilbert transform, does not coincide with ...
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26 views

Littlewood-Paley theorem at endpoints

Littlewood-Paley theorem says that the $L^p$ norm of the square function associated with $f$ is equivalent to the $L^p$ of $f$ when $p\in (1,\infty)$. I'm interested in the endpoint case. Why it ...
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41 views

Example of a compactly supported Lipschitz function with non-Lipschitz Hilbert transform

Suppose $f$ is a compactly supported measurable function (say in the interval $[-1,1]$) which is Hölder continuous of order $\alpha\in (0,1)$. I have read that the Hilbert transform $Hf$ of $f$ is ...
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23 views

How can large u(1/2)?

Actually it is rudin's exercise 11.13. Let $u$ is positive harmonic on $U$, unit disc and $u(0)=1$. How can large $u(1/2)$ be? What I tried is this; By theorem 11.30 that every positive harmonic ...
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44 views

Calderon-Zygmund Operator Associated to Zero Kernel

We say that a Calderon-Zygmund operator (CZO) $T:L^{2}(\mathbb{R}^{n})\rightarrow L^{2}(\mathbb{R}^{n})$ (i.e. a bounded linear operator) is associated to a CZ kernel ...
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16 views

Solving for values of a Fourier transform

Let $f(x)$ be an integrable function on $\bf R$. Given constants $A,B>0$, does there always exist an $x$ satisfying the following equation? $$A=Bx+\hat{f}(x)=Bx+\int^\infty_{-\infty}f(t)e^{ixt}dt$$ ...
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Expected value formula holds for family of functions.

Does Dynkin's formula hold for any function $f: \mathbb{R}^d \to \mathbb{R}$ such that $f$ and all of its partial derivatives of order $\le 2$ have at most polynomial growth at $\infty$?
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39 views

The distribution solution to $L^{+} u=0$ is u=0, where $L^{+}=-\frac{d}{dx}+x$?

Consider the creation operator $L^{+}=-\frac{d}{dx}+x$. If $u\in L^2(\mathbb{R})$ and is a distribution solution to the equation $L^{+}u=0$, then for any $\phi\in C_0^{\infty}(\mathbb{R})$ we have ...
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23 views

Computing Haar Measure

How do we compute both (Left and Right) Haar measure on the following group $$G = \left\{ \begin{bmatrix}x & y \\ 0 & 1 \end{bmatrix} \bigg| \ x\in \mathbb R \ \ \text{and }\ \ y \in ...
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87 views

Reference for Harmonic Analysis?

I'm looking primarily for references for Harmonic Analysis. I'm mostly considering Doran&Fell or Deitmar, but I have access to lectures using Stein as well. The important thing is covering ...
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18 views

Fourier Transform of Distribution Equal to the Distribution Itself

We define $T(t)=a\delta^{(n)}+bt^n$ for $a, b$ nonzero complex constants and $n$ a nonnegative integer. I want to find the combinations of $a, b, n$ such that the fourier transform of $T$ is equal to ...
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22 views

*-representations, unitary representations, and adjunctions

I've been reading Folland's Abstract Harmonic Analysis, and I am currently in the section on the correspondence between unitary representations of a locally compact group $G$ and $*$-representations ...
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29 views

Haar measure on $\mathbb{C} \setminus 0$.

I want to construct a Haar measure on $\mathbb{C} \setminus 0$. That is, a Borel measure $\mu$ on $\mathbb{C} \setminus 0$ such that $\mu(zS) = \mu(S)$ for all $z \in \mathbb{C} \setminus 0$ and all ...
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29 views

A question on maximal operator in Stein's real variable methods, orthogonality and oscillatory integrals.

On page 111 of Stein the maximal function defined as $\mathcal M_0 f(x) = \sup_{t>0} |f*\Phi_t(x)|$ (on page 106), where $\Phi$ is a smooth function supported in the unit ball about the origin with ...
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21 views

Some properties of the homogeneous spaces on $\mathbb{T}$

I am reading the first chapter of Y. Katznelson "An introduction to harmonic analysis". There is a definition of homogeneous space $B$ on $\mathbb{T}$, and then it is proved that the trigonometric ...
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14 views

Invariant subspaces of $L^{\infty}(G)$

I am reading a book named "Lectures on Amenability". The author Dr. Volker Runde defines the following : Let $G$ be a locally compact group, and let $E$ be a subspace of $L^{\infty}(G)$ containing ...
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59 views

BMO space and log-lipschitz regularity

Let $\Omega$ an open bounded domain in $R^n$ with smooth boundary and consider a function $u \in W^{1,p}(\Omega) (2<p < +\infty)$ . Suppose that for every ball $B \subset \subset \Omega$ exists ...
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(Exponential) Growth of Operator Norm of Uncentered Maximal Function

Define the uncentered Hardy-Littlewood maximal operator $M$ by $$Mf(x):=\sup_{x\in B}\dfrac{1}{\left|B\right|}\int_{B}\left|f\right|,$$ where we the supremum is taken over all (open) balls $B$ ...
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28 views

Counterexample to $L^1$-boundedness of the maximal operator $f \mapsto f^\#$ with $f^{\sharp}(x):=\sup_{Q\ni x}\frac{1}{|Q|}\int_{Q}|f-(f)_{Q}|dy$

My question concerns the sharp maximal operator, mapping a locally integrable function $f\colon\mathbb{R}^{n}\to\mathbb{R}$ to $f^{\sharp}\colon \mathbb{R}^{n}\to\mathbb{R}$, where ...
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50 views

Proving the Fourier inversion formula in finite abelian groups

The following exercise is exercise 2.2.3 from these lecture notes by Daniel Bump. If $G$ is a finite abelian group, then denote by $G^{\ast}$ the set of its characters, further if $x \in G$ let ...
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31 views

Fourier inversion formula on finite abelian groups

The following exercise is exercise 2.2.3 from these lecture notes by Daniel Bump. Let $\mathcal F : L^2(G) \to L^2(G^{\ast})$ be the Fourier transform, defined by $\mathcal{F}f = \hat f$, where ...
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65 views

Clarification on something in “Harmonic Analysis - real variable methods, orthogonality and oscillatory integrals” by Elias Stein.

On page 97 in the book, how did Elias inferred that instead of the factor $(1+2^k|y|/t)^N$ we must instead insert the factor: $\frac{t^L (\epsilon+ 2^{-k}t+\epsilon ...
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37 views

Fourier analysis on groups, and the isomorphism of characters in the “classical” setting

I am reading these lecture notes By Daniel Bump about Character Theory on Abelian Groups. If $G$ is a group, then $G^*$ denotes its characters, the set of homomorphisms $\pi : G \to \mathbb ...
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16 views

Abel transform of $R_z(\Delta)$

The spectrum of the Laplace operator $\Delta$, as self-adjoint unbounded operator on $L^2$ is equal $]-\infty ,0]$. The resolvent $R_z$ is defined for $z \in \mathbb C \setminus ]-\infty ,0] $. Using ...
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42 views

Underdamped free vibration proof

I need to prove the solution form of: $$y''+2cwy'+wy=0$$ My book says, after assuming a solution of the form $Ce^pt$, you can show that: $$y=[A\sin(wt)+B\sin(sw)] \cdot e^pt$$ I tried using the ...
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34 views

Spherical Fourier Transform of $p_t$

Let $p_t$ denote the heat kernel of the Laplacian $\Delta = \sum_{i=1}^{n}\frac{d^2}{dx_{i}^2}$ on $\mathbb R^n$, I want compute the Spherical Fourier Transform of $p_t$ on $\mathbb R^n$ ? Thank you ...
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Cylindrical boundary condition

There is a cylinder with radius $\rho_o$ and height $h$. The lids are on the planes $z=0$ and $z=h$. $\nabla ^2 \phi = 0$ , $\phi = \phi_o$ on the upper lid, $\phi=0$ every where else on the ...
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36 views

Wavelet through the lens of Harmonic analysis

I need some references for starting to study about wavelet. I have enough information about abstract Harmonic analysis. Thanks!
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45 views

Harmonic Oscillators: Differential Equations

The book being used for this course is Differential Equations, Dynamical Systems, and an Introduction to Chaos by Morris W. Hirsch. The question is as follows. Suppose there are two masses $m_1$ ...
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49 views

Can a lower semi-continuous function blow up to $+\infty$?

I was confused in lecture today that the professor said a l.s.c function can not blow up to $+\infty$... In my point of view, a l.s.c function can blow up to $+\infty$... for example, we take ...
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57 views

Is this function bounded below?

Let a bounded open set $\Omega\subset \mathbb R^N$ be given. Let $f$: $\Omega\to (0,+\infty]$ be given and we assume $f$ is locally integrable and there exists a constant $C>0$ so that $$ ...
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Asymptotic behavior of $\sum_{t=1}^T \frac{s_t}{t}$, when $\sum_{i=1}^T s_i = A$

consider the harmonic series: $$ h_T = \sum_{t=1}^T \frac{1}{t} $$ We know that $h_T \in O(\log T)$. Now consider a modified version of the harmonic series: $$ g_T = \sum_{t=1}^T \frac{s_t}{t} $$ ...
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41 views

What is the Fourier Transform of an absolute function?

I would like express that the Fourier transform of the function $$ |f(x)| $$ as $$ \widehat{|f|}(\xi) = \text{a function of } \widehat{f}(\xi) $$ In fact, I want to know the relation of ...
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Left invariance of a differential operator

Given $x, a\in G$ with G a group and x and a fixed, does the left invariance of a differential operator D on G imply that $D[f(a^{-1}x)]=D[a^{-1}f(x)]$?
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102 views

Derive Hausdorff-Young inequality from Paley's inequality

Given a sequence $(c_j)_{j\in\mathbb{Z}}$ of complex numbers with $\lim_{|j|\to\infty}c_j=0$. Define the rearrangement $c_j^*$ as follows: for $j\geq 0$, $c_j^*$ is the $j+1-$th largest element of the ...
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70 views

Structure of $\Bbb Q_p/\Bbb Q$

The group $\Bbb A_\Bbb Q/\Bbb Q$ of adeles mod rationals is: Isomorphic to the solenoid Isomorphic to the group $\Bbb A_\Bbb Z/\Bbb Z$ Dual to the rational numbers This raises the question of the ...
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Inverse limit of $\Bbb Q/q\Bbb Z$ isomorphic to finite adeles?

Let $\Bbb Q/q\Bbb Z$, for some positive rational $q$, denote the quotient group of the discrete rationals by the subgroup of integers times $q$. For any $q_1, q_2 \in \Bbb Q^+$ and $n \in \Bbb N^+$ ...
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61 views

Are singular integral operators bounded on $L\log L$?

My question is regarding singular integrals of Calderon Zygmund type. It is known that the maximal function is bounded on $L\log L \mapsto L^1$ (but not on $L^1$) and satisfies the same operator ...
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The equivalent of “tempered distributions” for the Mellin transform?

The Fourier transform is defined for tempered distributions. For these distributions, the test functions are those functions decreasing more quickly at $\pm \infty$ than $|x|^{-n}$ for all n. In ...
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Do combined waves with non-rational frequencies have a common period?

I am facing a problem where I have two waves combined: \begin{equation} y = A\sin(b_1x)+B\cos(b_2x) \end{equation} Where $ b_1 $ and $ b_2 $ are non-rationals. i.e. \begin{align} & b_1 = ...
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What do we call the harmonics in a discrete Fourier series representation?

In harmonic analysis using discrete Fourier series, if I'm using the 0f, 1f, 2f, 3f and 4f for representation where f = frequency, what is the correct way to say how many harmonics I'm using for ...
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Unital amenable Banach algebras which is a proper two sided ideal in its second dual

I need some examples of "unital amenable Banach algebras which is a proper two sided ideal in its second dual".
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44 views

proof that the Fourier series of $ f\ast g $ uniformly converge.

Let   $f,g$  be   $2\pi$-periodic piecewise continuous functions. proof that the Fourier series of $ f\ast g $ uniformly converge. Where $ f\ast g $ denotes the convolution operator ...
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40 views

Verification of a weighted inequality calculation

I was reading Fourier analysis by J. Duoandikoetxea, and checking out the proof of the $(L^p,L^p)$ inequality $$ \left| \left| \left( \sum_j|T_jf_j|^2 \right)^{\frac{1}{2}} \right| \right|_p\leq ...
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43 views

Decide if the improper integral of a Fourier transform converges

I have the function: $$f(x)=\left\{\begin{matrix} e^{-x^{10}} & ,x>0\\ -e^{-x^{10}} & ,x<0 \end{matrix}\right.$$ I need to answer: ...
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56 views

Convergence of improper integral over Fourier transform.

So I have the Fourier transform $$ \widehat{f}(\omega)=\frac{1}{1+|\omega|} $$ of some function $f(x)$. I need to know if the two integrals below converge or not. $$ ...
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60 views

Solving integral equation using convolution and Fourier transform.

So I have the integral equation : $$\int_{-\pi}^{\pi} f(t)f(x-t) dt = -\cos (x).$$ I know that I should use Fourier transform or Laplace transform and to use the convolution theorem, but I'm not ...
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47 views

Argument for extenstion of the Fourier transform

Could anyone please point out if there is any mistakes in the following arguments for the extension of the Fourier transform from $\mathcal{S}(\mathbb{R})$ to $L^2(\mathbb{R})$: "Since the compactly ...
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40 views

$\||u|\|_{H^s(\mathbb R^n)} \le C \| u \|_{H^s(\mathbb R^n)}$ holds for even if $s$ is not an integer?

Let $u: \mathbb R^n \ni x\mapsto u(x) \in \mathbb C.$ I would like to know that the inequality $$ \||u|\|_{\dot H^s(\mathbb R^n)} \le C \| u \|_{\dot H^s(\mathbb R^n)} $$ or $$ \||u|\|_{H^s(\mathbb ...