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-...

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Decay property of oscillatory integrals in $\mathbb{R}^n$

We know that an oscillatory integral in $n$ dimensions is an integral of the form \begin{equation*} I(\lambda)=\int_{\mathbb{R}^n}e^{i\lambda\phi(x)}f(x)dx \end{equation*} where $\phi\in C^{\infty}$ ...
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1answer
37 views

Concluding that if $f\in L^{1}_{\text{loc}}(\mathbb{R}^{n})$ and $\mathcal{M}(f)\in L^{1}(\mathbb{R}^{n})$ then $f=0$ a.e.

Let $\mu$ be a Lebesgue measure, $\mathcal{M}$ be the Hardy-Littlewood Centered Maximal Function, and the rest as in the title. Then to this end I define $f_{R}(x):=f(x)\chi_{|x|\le R}$ where $\chi$ ...
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141 views

Folland real analysis 9.11

This comes from question 9.11 of Folland's Real analysis textbook. Unfortunately, I have no idea to how to start with this question. So can some one help me with part $a$? For part $a$, I can not ...
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Is set of adjoint derivations on a Banach algebra to Banach bi-module closed subspace?

Let $A$ be a Banach algebra and $B$ be a Banach space. Then $B$ is called left( resp. right) $A$- module if there is continuous representation(resp. anti-representation) $T : A \to BL(B)$ (Bounded ...
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65 views

Fourier transform of a Gaussian

I am trying to solve the following exercize: Show that Fourier transform of a Gaussian (a function of the form $Ae^{-\frac{x^2}{\sigma^2}}$) is also a Gaussian. So I did the required calculation (I ...
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Question on the existence of the limit in S. Petermichl's paper

I have read S.Petermichl's paper Dyadic shifts and a logarithmic estimate for Hankel operators with matrix symbol many days and got trouble in the proof of Lemma 2.1 in this paper. Let us see Lemma 2....
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11 views

Inverse transform of digamma function

Let $g$ be in $C_c^\infty(\bf R)$, and $h$ its Fourier transform. There is the following equation: \begin{align}\frac{1}{2\pi}\int^\infty_{-\infty}h(r)\frac{\Gamma'}{\Gamma}(1+ir)dr=&-\gamma g(0)+...
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35 views

Derive mean value property of harmonic function using this method

We can view harmonic function $u(x,y,z)$ as a wave function that does not depend on time $t$. Let $\overline{u}(r)$ be its average value on the circle with radius $r$. Assume we have proved that it ...
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42 views

Is this a right calculation of Fourier transform?

I am trying to solve: Calculate Fourier transform (on $-\infty < x < \infty $) of: $$ f(x) = \begin{cases} 1, & -L<x<L \\ 0, & |x|\geq L \end{cases} $$ My ...
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60 views

Poisson summation in analytic number theory, an example

I've read a theorem of a course of analytical methods in theory of numbers, and I want to ask to clarify it, since I know this theorem from other context. The theorem is Poisson summation formula (I ...
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13 views

Minimal error given when making an approximation of $f(x)$ by sines and cosines

I am studying by myself Fourier analysis and have encountered the following problem: We are trying to approximate a function by a finite sum of sines and cosines with general constant coeficients: $$...
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11 views

wavlete transform vs (scaled) Gabor transform

I've read about the scaled Gabor transform $$(G_\Psi f)(b,a)(\omega) = \frac{1}{\sqrt{a}} \int_\mathbb{R} f(x)\Psi(\frac{x-b}{a})e^{-i\omega x}dx$$ and the wavlete transform $$(L_\Psi f)(b,a) = \frac{...
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39 views

Question while reading “Harmonic analysis as the exploitation of symmetry — a historical survey”: what is the meaning of “identity map”?

I am reading through the following article: Mackey, George W. Harmonic analysis as the exploitation of symmetry—a historical survey. Bull. Amer. Math. Soc. 3 (1980), no. 1, part 1, 543–698 This is a ...
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22 views

Prove that the special Hermite functions are eigenfunctions of $R_{x,y}$?

How to prove that the special Hermite functions are eigenfunctions of the rotation operators $$R_{x,y} = x\frac{\partial}{\partial y}- y\frac{\partial}{\partial x}.$$ Where the special Hermite ...
4
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2answers
46 views

Example of inverse semigroup with at least two idempotent elements

We say that the semigroup $S$ is inverse semigroup if for any $s\in S$ there is a unique $t\in S$ such that $sts=s,\ tst=t$. Suppose that $E(S)=\{e:\ e\in S,\ e^2=e\}$ and define $$s\sim t\...
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3answers
61 views

Riemann Lebesgue Lemma application? [closed]

Riemann Lebesgue Lemma shows that if $f \in L^1 ( \bf R)$ then the Fourier transform of $f$ goes to $0$. Does this also implies that $f(x) \to 0$ as $\vert x \vert \to \infty$ ?
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43 views

Homogeneous distribution

In Wikipedia, it says The Dirac delta function is homogeneous of degree −1, with the following formula: However, I can not understand why the last equality is true. Can someone show me the detailed ...
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28 views

Do we have for $\lim_{r\to \infty}\frac{1}{rlog(r)}(\int_0^{2\pi}u(re^{i\theta})d\theta)$ exists for $u$ subharmonic?

Let $u:\mathbb{C}\to \mathbb{R}$ be a subharmonic function. Do we have that the limit $\lim_{r\to \infty}\frac{1}{rlog(r)}(\int_0^{2\pi}u(re^{i\theta})d\theta)$ converges to a (possibly infinite) ...
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124 views

Can the real part of an entire function be bounded above by a polynomial?

Let $f:\mathbb{C}\to \mathbb{C}$ be an entire function such that $Re(f)\le |p(z)|$ for some polynomial, can we derive that $f(z)$ is a polynomial. If $p(z)$ is constant, then this can be shown by ...
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57 views

references for abstract harmonic analysis

I am currently studying a basic course in abstract harmonic analysis. I do have a good background in abstract algebra and functional analysis but I have not done a course in Fourier analysis. Is it ...
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47 views

Strichartz Estimate with Fourier Transform

Let $f$ be a Schwartz function. Prove that, whenever $2\le r < \infty,$ $$\| e^{it \Delta} f\|_{L^{3r}(\mathbb{R}^2_{xt})} \le c \| \widehat{f}\|_{r'},$$ Where $1/r + 1/r' = 1.$ My Attempt My ...
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Some problems about symmetric convolution semigroup on the unit circle

These are problems from Example 1.4.2 of Fukushima's book "Dirichlet forms and symmetric Markov processes". Let $\Lambda$ be the set of all real sequences $\left\{\lambda_n\right\}_{n\in\mathbf{Z}}$ ...
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24 views

Multiplier operators on anisotropic weighted $L^2$ spaces

Suppose $\mathcal{M}$ is a multiplier operator on $L^2(\mathbb{R})$, in the sense that, for any $u(x)\in L^2(\mathbb{R})$, $\widehat{\mathcal{M}u}(k)=m(k)\hat{u}(k),$ where the scalar complex function ...
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1answer
45 views

$L^{p}$ identity for the maximal operator.

This is probably a very easy question, but I'm failing to understand it. Given a function $f \in L^{p}(\mathbb{R}^n)$, $1<p\leq \infty$, we define the uncentered maximal function of $f$ as $$ \...
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1answer
30 views

Number of copies of irreducible unitary representation in $L^2(G)$ for compact group $G$?

Peter-Weyl Theorem is concerned with expressing $L^2(G)$ as closure of direct sum of subspace generated by irreducible unitary representation. Every irreducible representation on a compact group is ...
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37 views

Examples of non unimodular groups

I'm looking for criteria to know if a (lie) group is unimodular, the thing is that I only know one non unimodualr group that arrises naturaly, the affine group ax+b, and their direct generalizations. ...
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21 views

Does this transformation(harmonic analysis) exist?

Assuming that there are two disjoint sets(A and B) of high dimensional $N^d$ integers. Each point $V$ can be expressed by a periodic function: $f(t)=v_1*cos(t) + v_2*sin(2t) + v_3*cos(2t)+v_4*sin(4t)+...
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26 views

Harmonic Motion - Fourier Approximation What does this mean below?

There is a method to solve systems under harmonic loading, harmonic balance method, which is basically obtaining fourier expansions of unknown response quantities and solving for coefficients of ...
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75 views

Estimate of a Fourier Multiplier Operator

Let $m_t (\xi) = \cos (2\pi |\xi| t).$ Define the operators, for $t>0,$ $$ T_t f = ( m_t \widehat{f} )^{\vee}.$$ It is asked to prove that, whenever $f$ is sufficiently regular, $$ \| T_t f\|_{\...
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20 views

Commutator Characterization of $BMO(\mathbb{R})$

Let $a:\mathbb{R}\rightarrow\mathbb{C}$ be a locally integrable function, and let $H$ denote the Hilbert transform. Suppose that the commutator operator $[a,H]$ defined by $[a,H]f:=aH(f)-H(af)$ is ...
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1answer
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Invariance of noneuclidean laplacian

In a book I'm reading it says: Putting $f(x,y)=F(u,v)$ with $\gamma(x+yi)=u+iv$ and using Cauchy-Riemann equation for $\gamma(z)$, we have $$\Delta f(x,y) \overset{(*)}=-y^2(u_x^2+v_x^2)(F_{uu}+F_{vv}...
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1answer
38 views

Non-compactness of support of linear KdV equation solution

The last question in Linares and Ponce's 'Introduction to Nonlinear Dispersive Equations's first chapter asks the reader to prove that, if the following IVP is given: $$\begin{cases} \partial_t u + \...
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1answer
35 views

How to prove that $f(x) = \int_{0}^{\infty} f_{\lambda} (x) \, d\lambda$ where $f_{\lambda} $ is the eigenfunctions of $\Delta$

On Euclidean space $\mathbb R^n$, how to prove that $$f(x) = \int_{0}^{\infty} f_{\lambda} (x) \, d\lambda ,$$ where $\Delta f_{\lambda} (x) = -\lambda^2 f_{\lambda}(x) $, whith $\Delta$ is the ...
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Laplacian on the sphere (proof of a proposition)

$$ Let \;f:U \to \mathbb{R}\;be\;a\;function\;on\;an\;open\;subset\;of\;S^{m-1}\;which\;is\;the\;restriction\;of\;a\;smooth\;function\;F:U{'} \to \mathbb{R}\;defined\;on\;an\;open\;subset\;of\;{{\...
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A question about general Marcinkiewicz interpolation theorem

The general Marcinkiewicz interpolation theorem states as following: If $T$ is a linear operator of weak type $(p_0,q_0)$ and of weak type $(p_1,q_1)$ where $q_0\neq q_1$, then for each $\theta\...
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2answers
62 views

Questions about delta function.

Let $z \neq 1$ be a complex number. Then \begin{align} \frac{1}{1-z} = \sum_{n=0}^{\infty} z^n. \end{align} We have \begin{align} \frac{z^{-1}}{1-z^{-1}} = \sum_{n=1}^{\infty} z^{-n}. \end{align} ...
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74 views

Scale resolution and frequency resolution in continuous wavelet transform

I'm reading the well known wavelets tutorial by Robi Polikar here. In part 3, about figure 3.7 and 3.8, it says "lower scales (higher frequencies) have better scale resolution (narrower in scale, ...
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30 views

Proving Div-Curl Lemma through Paraproducts

I want to to prove the classical div-curl lemma of Coifman-Lions-Meyer-Semmes in the following form: Div-Curl Lemma. Let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ be a Schwartz function such ...
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Eigenvalues for correlation matrix which have the form of an harmonic function

As a continuation to this question, I took the matrix $C_{2 \times 2}$ which is: $$C=\left[ \begin{array}{} a& ace^{-\frac{|\phi_1-\phi_2|}{2}}\\ ace^{-\frac{|\phi_1-\phi_2|}{2}} &...
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35 views

Fourier series calculation

I had an exam question today that stated something along the lines of the following: "Let $f$ be an even function given by $f(x)=x$ on $[0,\pi]$ and extend $f$ to $\mathbb{R}$ by $2\pi$-periodicity. ...
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How come $ce^{-\frac{\phi_i-\phi_j}{\rho}}$ has the form of $e^{-in\phi_i}$ and how to calculate it's eigenvalues

How to prove Euler's Formula $e^{i\theta} = (cos\theta + isin\theta)$? I know this is kind of basic and I am familiar with this equality for a long time. But, how do I prove it? And another question:...
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1answer
199 views

Prove the uniqueness of poisson equation with robin boundary condition

We have $\Delta u=f$ in $D$, and $\dfrac{\partial u}{\partial n}+au=h$ on boundary of D, where $D$ is a domain in three dimension and $a$ is a positive constant. $\dfrac{\partial u}{\partial n}=\...
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17 views

Laplacian of a radial function on a riemannienne symmetric space

I would like to know : Is that the Laplacian $L$ of any radial function $f$ on a riemannienne symmetric space $X$ is a radial function? Thank you in advance.
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The decay rate of Hormander lemma is optimal or not?

The Hormander lemma about oscillatory integral operators states that $T_\lambda f(x)=\int e^{i\lambda S(x,y)}a(x,y)f(y)dy$, while the Hessian of $S(x,y)$ is nondegenerate, then $||T_\lambda||_2 \leq C\...
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15 views

How does the weighted superposition of irreps make the Fourier transform of a finite group unitary?

This is supplementary to this question. In the lecture note of Andrew Childs on Nonabelian Fourier analysis, it is said that the Fourier transform of a finite group is the weighted superposition of ...
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1answer
39 views

Why is the Fourier transform of a non-Abelian finite group the weighted superposition over all irreps?

I am going through the lecture note of Andrew Childs on Nonabelian Fourier analysis. I would like to quote from the note: My question: Why does it have to be weighted superposition and not equal ...
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1answer
31 views

On a property of $(Mf)^{\delta}$

For each positive $C$, define a set $$A_C=\left\{g\ge0: \frac{1}{|I|}\int_Ig\le C\inf_{x\in I}g(x) \text{ for any interval } I\right\}$$ In other words, elements in $A_C$ are non-negative functions ...
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1answer
48 views

The support of $f(x)= \cos(x)$

The support of a function is the closure of the set of points where the function has non zero values. The function $f(x)=\cos(x)$ is zero only at the points $x=\frac{(2k+1)\pi}{2}$, $k \in \mathbb{Z}...
2
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1answer
40 views

Gradient Estimate for Logarithmic Cutoff Function

In Notes 4 on regularity of harmonic maps, T. Tao asserts the following lemma, which in short, allows us to localize a map $u:\Omega\rightarrow S^{m-1}$ (here, $\Omega\subset\mathbb{R}^{2}$ is a ...
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37 views

Strong period of self-convolution of a strongly periodic cyclic function

Let $f : \mathbb{Z}_N \to F$ be a function with a $F$ a field. We say that $f$ has maximum period if the smallest positive integer $r$ (with $r \mid N$) such that $f(j) = f(j + r)$ for all $j \in \...