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, ...
4
votes
1answer
41 views
Proof of weak maximum principle.
I read a proof in my book on pde which I find a bit strange. Let $\Omega$ be some bounded domain. For $f\in\mathscr C^2(\Omega)\cap\mathscr C^0(\Omega)$ satisfying $\Delta f\geq0$ it holds that ...
4
votes
1answer
80 views
Example of an unbounded operator
Can somebody give me an easy example of a linear operator which maps $L^1(\mathbb{R}^n)$ to $L^1(\mathbb{R}^n)$ and $L^\infty(\mathbb{R}^n)$ to $L^\infty(\mathbb{R}^n)$ (but not boundedly) but does ...
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votes
0answers
21 views
Constraints/Initial of second order differential equation?
Let
$$\Psi(x)=\sum_{k=-j}^j \psi(x,k,j)=\sum_{k=-j}^j \frac{2 \pi}{j}\,e^{(i\,2\pi\,(k-1)\,x / j)} \qquad(1)$$
with $i,j,k\in \Bbb N$
and
$$\frac{\partial^2\psi}{\partial x^2}=\psi(x)+ x ...
2
votes
1answer
27 views
Carleson embedding theorem
The Carleson embedding theorem gives a criterium by which to decide when for a measure $\mu$ the operator that takes a function on the real line to its harmonic extension (by convolution with Poisson ...
1
vote
1answer
42 views
Proof that $dx/|x|$ is a Haar measure on non-zero reals?
Most importantly, what is the meaning of this notation $\lambda = dx/|x|$? How do I compute say $\lambda(0,1)$ for example?
2
votes
1answer
49 views
Weak $L^{p}$ spaces are quasi-normed?
Let $(X,\mathcal{B}, \mu)$ be a measure space. Then for $0< p < \infty$ by definition
$L^{p,\infty}(X,\mathcal{B}, \mu)$ is the class of all measureable functions $f$ such that
...
2
votes
1answer
24 views
Showing atomic $H^{1,p}$ is a Banach Space
Consider $1<p\leq\infty$ and let $H^{1,p}=H^{1,p}(\mathbb{R}^n)$ be an atomic Hardy space, that is $H^{1,p}\subset L^1(\mathbb{R}^n)$, and $f\in H^{1,p}$ if there exists a sequence of $p$-atoms ...
6
votes
2answers
84 views
Good book on topological group theory?
I'm looking for a good introduction to the theory of locally compact groups and their representations. It may assume the reader to be familiar with basic group theory, topology and measure theory.
2
votes
1answer
161 views
Fourier analysis questions
Can anyone give me a hand with the proof of this properties?
Prove that:
a) The linear span of the set $\left\{T_bh/b\in\mathbb{R}\right\}$ is dense in $L_2(\mathbb{R})$, where $h(x)=e^{-\pi x^2}$. ...
3
votes
1answer
34 views
Are the continuous functions on $G$ dense in $L^{1}(G)$?
If $G$ is a locally compact group, is the set $C_{c}(G)$ of all continuous functions on $G$ with compact support dense in $L^{1}(G)$?
3
votes
0answers
34 views
The Hilbert transform and function spaces
I'm trying to find some concrete details about what the image set is for the Hilbert transform of different function spaces. In particular, I'm interested in what function spaces emerge when the ...
3
votes
1answer
40 views
Why are Haar measures finite on compact sets?
I'm working through the answer by t.b. to another user's question here:
A net version of dominated convergence?
because I am trying to work through a related problem and I think it will be ...
3
votes
0answers
22 views
Are Haar measures complete?
If $G$ is a locally compact group and $\mu$ is a left Haar measure for $G$, then is the measure space $(G,B(G),\mu)$ complete (where $B(G)$ is the set of Borel subsets of $G$)?
Or do we have to take ...
1
vote
1answer
76 views
Why is logarithm in BMO
I read that $\log|x|$ is supposedly a typical example of a BMO function. How do I see that it is in BMO actually?
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0answers
17 views
Convergence of an Integral in a locally compact group
I'm trying to finish an exercise which I asked about earlier here:
Mapping $G$ into its group algebra as left multiplication. Continuous?
$\bf{\text{The setting:}}$
Let $G$ be a locally compact ...
0
votes
0answers
17 views
Bounding the partial derivatives of $\left(\dfrac{1+ |\xi + \tau|^2}{1 + |\xi|^2}\right)^{s/2}$.
Let $\tau = (\tau_1,\ldots,\tau_n) \in \mathbf{R}^n$, $s\geq 0$, and define the function $f : \mathbf{R}^n \to \mathbf{R}$ by
$$
f(\xi) = \left(\dfrac{1+ |\xi + \tau|^2}{1 + |\xi|^2}\right)^{s/2}.
$$
...
4
votes
1answer
73 views
Involution in $L^{1}(G)$ is isometric.
(Sorry for asking so many questions of the same type. There is an underlying issue that I think once resolved will allow me to understand them all at once.)
Let $G$ be a locally compact group, and ...
2
votes
1answer
37 views
How to use the modular function of a locally compact group?
Let $G$ be a locally compact group with left Haar measure $\mu$.
Let $\delta:G\to(0,\infty)$ the unique modular function such that $\delta(x)\mu(E) = \mu(Ex)$ for all Borel sets $E$ and $x\in G$.
...
1
vote
3answers
65 views
Introduction to Abstract Harmonic Analysis, reference suggestions?
I'm looking for a good starting book on the subject which only assumes standard undergraduate background.
In particular, I need to gain some confidence working with properties of Haar measures, so I ...
3
votes
0answers
48 views
Mapping $G$ into its group algebra as left multiplication. Continuous?
I am reading an appendix on Group algebras which contains the following Proposition which I am trying to prove:
Proposition: Let $G$ be a locally compact group, with $\zeta\in L^{p}(G)$ fixed. ...
2
votes
2answers
59 views
An integral$\frac{1}{2\pi}\int_0^{2\pi}\log|\exp(i\theta)-a|\text{d}\theta=0$ which I can calculate but can't understand it.
I have encountered this integral:$$\frac{1}{2\pi}\int_0^{2\pi}\log|\exp(i\theta)-a|\text{d}\theta=0$$ where $|a|<1$ in proving Jensen's formula. because I am stupid, I don't know why it is equal ...
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votes
0answers
42 views
Higher dimensional integration by parts, specific case.
Suppose $\sigma$ is $C^\infty$ function on $(\mathbf{R}^n)^2\backslash \{(0,0)\}$ that is supported in the cube $[-1/4,1/4]^{2n}$ which satisfies
$$
|\partial_{y_1}^{\alpha_1}\partial_{y_2}^{\alpha_2} ...
1
vote
3answers
44 views
Fourier transform of a compactly supported function
In which space does the Fourier transform of a smooth compactly supported function $\phi$ lie?
I would not say it lies in $\mathcal{S}$, heuristically as one can approximate the step function which is ...
0
votes
1answer
21 views
Characterisation of the spectrum of certain unitary representations on $L^2(G)$
I have been informed of the existence of theorems in harmonic analysis that will allow me to calculate the spectrum of given unitary operators $L^2(G)$ where $G$ is a locally compact group. So far I ...
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votes
0answers
22 views
Laplacian on Reductive coset spaces
Let us consider the sphere $S^n$ (embedded in $\mathbb{R}^{n+1}$) and let $X_i$ denote the vector fields which generate rotations about the $x_i$-axis. My questions are:
(a) Is it true that ...
3
votes
1answer
44 views
number of zeros of the superposition/interference of sine oscillations
There is a tricky problem to solve and we ask for your kind help.
In a rather simple version of the problem, we have a set of oscillations of type $f(x)=\sin (\omega x + \phi)$ with different phases ...
1
vote
1answer
49 views
Harmonic Measure & Brownian Excursion
I have a disk of radius R removed from the plane. Brownian excursions are emanated from infinity. I am tasked to find a probability that the brownian motion impacts an arc of the disk boundary. I am ...
2
votes
0answers
36 views
$H^s(\mathbb{T})$ space is a Banach algebra with pointwise product
I have ran across the following theorem but the given proof does not convince me.
Theorem Let $u, v \in H^s(\mathbb{T})$ with $s>1/2$. Then the pointwise product $uv$ is in $H^s(\mathbb{T})$ and ...
1
vote
1answer
25 views
Measure of the boundary of a union of cubes.
Suppose that we are given a collection of dyadic, mutually disjoint, open cubes in $\mathbf{R}^n$ in which the the union of all the cubes has finite measure. Is it necessary that the boundary of the ...
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votes
0answers
25 views
Harmonics of a rectangular cuboid [closed]
What are the harmonics, or normal modes, of a rectangular cuboid (brick, or sheet, if it's thin)? Which material constants?
7
votes
2answers
117 views
Mean Value Property of Harmonic Function on a Square
A friend of mine presented me the following problem a couple days ago:
Let $S$ in $\mathbb{R}^2$ be a square and $u$ a continuous harmonic function on the closure of $S$. Show that the average of ...
1
vote
0answers
26 views
DFT in complex form and Trigonometric Polynomial Interpolation: why different dimensions of basis vectors set?
I'm trying to figure out, how coefficients of Discrete Fourier Transform in complex form are converted into coefficients of Trigonometric Polynomials Interpolation:
Say, I have a function vector with ...
0
votes
1answer
15 views
An approximation of lengths dealing with elements of cubes. Show $|x-y| \approx \ell(Q) + |x - c_Q|$.
So the problem I have at hand should be rather elementary, but I can't figure out. It's in a proof I'm trying to understand completely. Here's what I need to do exactly:
Show that $|x-y| \approx ...
1
vote
1answer
27 views
Extracting Harmonic series components
I have a number which is made up of a Harmonic series.
1/2 + 1/3 + 1/4 etc.
Some of the components may not be in the number..
1/2 + 1/7 + 1/11 etc.
Is it possible to recover the individual ...
2
votes
1answer
54 views
Why does this inequality for all characteristic functions imply it for simple functions?
This question is probably obvious, but I'm not seeing how to obtain it.
A simple function is said to be finitely simple if its support is of finite measure. Let $(X_1,\mu_1)$, $(X_2,\mu_2)$, and ...
0
votes
0answers
19 views
Limit of integrals of a sequence of locally integrable continuous functions.
Suppose that $\sigma$ is locally integrable and continuous on $(\mathbf{R}^n)^m$. Fix $\vec a \in (\mathbf{R}^n)^m$ and let $f_1,\ldots,f_m$ be Schwartz functions on $\mathbf{R}^n$. In the proof I'm ...
1
vote
1answer
45 views
Showing a Schwartz Function Bound
I have a question on how to get the correct upper bound of a Schwartz function. Unfortunately, I've never understood this even though I've seen my professors do it a thousand times. I figured it's ...
2
votes
1answer
40 views
Does $|T(f) - T(g)| \leq |T(f-g)|$ hold for a sublinear operator $T$?
Let $X,Y$ be function spaces with functions taking values in $\mathbf{C}$. An operator $T:X\to Y$ is called sublinear if for all $f,g \in X$ and all $\lambda \in \mathbf{C}$, we have
$$
|T(\lambda f)| ...
1
vote
1answer
40 views
Lorentz Space property: $\left\| f \right\|_{L^{q,s}} \leq \lim\limits_{n\to\infty} \| f_n \|_{L^{q,s}}$
I would like to understand a statement similar to Fatou's Lemma in the Lorentz space setting. It is as follows.
Suppose $0 < q,s < \infty$ and $f_n,f$ are measurable functions on a
...
3
votes
2answers
75 views
Identity involving partial sums of Fourier series
Suppose $f$ is a continuous periodic function and $S_Nf(x) = \sum^N_{n=−N} \hat f(n) e^{inx}$, where $$\hat f(n)= \frac{1}{2\pi}\int_0^{2\pi}f(x)e^{-inx} dx.$$
How can I show that ...
2
votes
1answer
25 views
Given $\widehat{\varphi} = \chi_{[-1/2,1/2]}$, show the system $\{\varphi(2^v x)\}_{v \in \mathbb{Z}}$ is not orthogonal
If $$\widehat{\varphi} = \chi_{[-\frac{1}{2},\frac{1}{2}]}$$ then it is not hard to compute via the inverse Fourier transform that $$\varphi(x) = \frac{\sin(\pi x)}{\pi x}$$ so we need to show ...
0
votes
1answer
67 views
Imaginary complex numbers
Let $z=x+iy$ and $v=2xy$, show that $v=Im[z^2]$ and find a harmonic conjugate of $v$ on domain $D$. Also find the largest domain $D$ on which $v$ is harmonic.
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0answers
51 views
Fourier algebra and multipliers of a finite discrete group
Let $G$ be a finite discrete group. We denote by $A(G)$ the Fourier algebra of $G$ and $M_{cb}A(G)$ the space of completely bounded multipliers of $A(G)$.
Is is true that $A(G)=M_{cb}A(G)$ ...
0
votes
1answer
56 views
Which spherical harmonic function will correspond to such a representation?
On Wiki there's a figure displaying "visual representations of the first few spherical harmonics."
I was wondering exactly which spherical harmonic function will generate a representation like this ...
1
vote
0answers
74 views
Another aspect of Heisenberg uncertainty principle
In fourier transformation theory, we have the Heisenberg uncertainty principle, i.e.
Suppose $\phi$ is a function in [Schwarz space][1] which satisfies the normalizing condition ...
6
votes
2answers
87 views
Recovering a group from its C*-algebras and group algebra
Let $G$ and $H$ be locally compact groups. Does anyone know the answers to these questions?
Is it true that:
if $C^*(G)$ and $C^*(H)$ are $*$-isomorphic, then $G\cong H$?
if $C_r^*(G)$ and ...
1
vote
0answers
32 views
How to show that $ν(z)$ is Carleson measure
If $ν(z)=|1+z|^ β dμ(z)$, $β\in \!\ R^-$. How to show that $ν(z)$ is Carleson?
I know that $ν$ is Carleson measure if $ν(Q_I)\leqq \!\ c.I$
But how to apply this?
0
votes
2answers
53 views
Harmonic function, existence of a constant
May i ask you for a little help about a problem with harmonic function? It seems to be not that difficult, in a way even intiutively obvious but i don't really know how to show this explicitly.
We ...
2
votes
1answer
81 views
Are nilpotent Lie groups unimodular?
The continuous homomorphism $\Delta:G \rightarrow \mathbb{R}^{\times}_+$ is defined by
\begin{equation*}
\int_G f(xy)dx = \Delta(y)\int_Gf(x)dx
\end{equation*}
where $dx$ is a left Haar measure on ...
1
vote
1answer
39 views
$\int_{-\infty}^{\infty}f '\bar{f}'+x^2 f\bar{f}dx\geq \int_{-\infty}^{\infty}|f|^2 dx$?
Suppose complex function $f$ in the Schwartz Space, its definition see http://en.wikipedia.org/wiki/Schwartz_space how can we argue that
$$\int_{-\infty}^{\infty}f '\bar{f}'+x^2 f\bar{f}dx\geq ...
