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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469 views
Do discontinuous harmonic functions exist?
A function, $u$, on $\mathbb R^n$ is normally said to be harmonic if $\Delta u=0$, where $\Delta$ is the Laplacian operator $\Delta=\sum_{i=1}^n\frac{\partial^2}{\partial x_i^2}$. So obviously, ...
3
votes
0answers
89 views
Curvatures of contours of solutions of 3d Poisson's equation
Let $f(x,y,z)$ be a complex function in a 3d euclidian space that fulfill the Poisson's equation
$$\frac{\partial^2}{\partial x^2} f + \frac{\partial^2}{\partial y^2} f + \frac{\partial^2}{\partial ...
3
votes
3answers
159 views
What is a “domain” in the maximum-minimum principle?
The maximum-minimum principle says that
A harmonic function on a domain cannot attain its maximum or its minimum unless it is constant.
Here is my question:
If we restrict our attention in
...
1
vote
1answer
136 views
On covering lemma and Calderón–Zygmund decomposition
I am working on something which needs to understand covering lemmas and Calderón–Zygmund decomposition. These type of lemmas are as in the following link
...
6
votes
1answer
498 views
Theorem of Steinhaus
The Steinhaus theorem says that if a set $A \subset \mathbb R^n$ is of positive inner Lebesgue measure then $\operatorname{int}{(A+A)} \neq \emptyset$. Is it true that also ...
4
votes
1answer
373 views
Reference request: Fourier and Fourier-Stieltjes algebras
I would like to learn the basic theory of Fourier algebras and Fourier-Stieltjes algebras. In particular, I want to know how these two objects are defined in the case of not necessarily abelian ...
7
votes
0answers
226 views
Motivation for abstract harmonic analysis
I am reading Folland's A Course in Abstract Harmonic Analysis and find this book extremely exciting.
However it seems Folland does not give many examples to illustrate the motivation behind much of ...
2
votes
1answer
128 views
Surjective endomorphism preserves Haar measure
How to prove the following statement:
Let $G$ be a compact topological group and let $m$ be the Haar measure on it. Let $\varphi$ be a continuous endomorphism of $G$ onto $G$, i.e., the map $\varphi$ ...
1
vote
1answer
124 views
On a duality Fefferman-Stein's inequality
Let $M$ be the Hardy-Littlewood maximal operator. In the book "Weighted norm inequalities and Related Topics" by Rubio de Francia and J. Cuerva, page 150, theorem 2.1.2 states as the following:
*For ...
2
votes
1answer
197 views
Applications of Young's convolution inequality
Recall that the convolution of two functions is given by
$$f*g(y)=\int f(x)g(y-x)dx.$$
The well known inequality known as Young's inequality, say that
$$\|f*g\|_r\leq\|f\|_p\cdot\|g\|_q $$
provided ...
2
votes
0answers
324 views
Spherical harmonics give all the irreducible representations of $SO(3)$?
It is mentioned in Wiki that the spaces $\mathcal{H}_{k}$ of spherical harmonics of degree $k$ give ALL the irreducible representations of $SO(3)$. Could anyone tell me where can I find the proof? ...
10
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1answer
242 views
Stone-Weierstrass implies Fourier expansion
To prove the existence of Fourier expansion, I have to solve the following exercise, which supposedly follows from the Stone-Weierstrass theorem:
Let $G$ be a compact abelian topological group ...
11
votes
1answer
327 views
Properties of Haar measure
Let $G$ be a locally compact group (but not discrete) and let $m$ be its left Haar measure. Is it true that $\forall \epsilon$ $\exists$ $C$ such that $C$ is a compact neighborhood of the identity and ...
3
votes
1answer
220 views
Left regular representation of $L^1(G)$ for a locally compact group $G$
Let $G$ be a locally compact group (not discrete) and let $L$ be the left regular representation of $A = L^1(G)$ on itself i.e. $L: A \to \mathcal{B}(A)$ where $L(f): A \to A$, $L(f)(g) = f*g$. I want ...
4
votes
0answers
76 views
Preduals of Banach spaces and in particular of $\text{BMO}(\mathbf R^d)$
In general the predual of a Banach space is not unique. If there are multiple ones must they be isomorphic?
More specifically is $H^1(\mathbf R^d)$ the only predual of $\text{BMO}(\mathbf R^d)$ or ...
13
votes
3answers
352 views
Rate of divergence for the series $\sum |\sin(n\theta) / n|$
In the following we consider the series
$$ S(N;\theta)= \sum_{n = 1}^{N} \left| \frac{\sin n\theta}{n} \right| $$
parametrized by $\theta$. It is well known that this series (taking the limit ...
1
vote
0answers
178 views
Proving the maximum principle for harmonic real valued functions in $\mathbb{R}^n$
Assuming the principle is stated as such:
Let $U\subset\mathbb{R}^n$ be a bounded domain and $u$ harmonic in $U$ such that $\sup_{x\in U}u(x)\leq A$ for some $A\in\mathbb{R}$. Then either $\forall ...
1
vote
1answer
76 views
Restriction and completion of Haar measure on $\mathbf{R} \times \mathbf{R_d}$ to Borel $\sigma$-algebra
Let's consider the measure space $(G, \mathfrak{M}, \mu)$, where $\mu$ is the Haar measure on topological group $G:=\mathbf{R} \times \mathbf{R_d}$, ($\mathbf{R}$ is the group of reals with the ...
1
vote
1answer
160 views
Example of a locally compact connected Abelian group with non-$\sigma$-finite measure
I look for an example of an Abelian locally compact topological group $G$ such that:
$G$ is connected and Haar measure on $G$ is not $\sigma$-finite and $\{0\} \times G \subset \mathbf{R} \times G$ ...
6
votes
0answers
283 views
Positive definite function zoo
A positive definite function $\varphi: G \rightarrow \mathbb{C}$ on a group $G$ is a function that arises as a coefficient of a unitary representation of $G$.
For a definition and discussion of ...
1
vote
2answers
366 views
Bounded linear operators that commute with translation
I'm trying to read Elias Stein's "Singular Integrals" book, and in the beginning of the second chapter, he states two results classifying bounded linear operators that commute (on $L^1$ and $L^2$ ...
2
votes
0answers
123 views
Steinhaus theorem in topological groups
Let $(G,\cdot)$ be a locally compact Abelian topological group with Haar measure. It is known that if $B$ is a measurable subset of $G$ of finite and positive Haar measure then $int(B \cdot B)\neq ...
2
votes
1answer
180 views
Asymptotic error of Fourier series partial sum of sawtooth function
In Iwaniec's book, Topics in Classical Automorphic Forms, pg. 4, he gives the statement:
$$\{x\}=\frac{1}{2}-\sum_{n=1}^N\frac{\sin 2\pi nx}{\pi n}+O((1+||x||N)^{-1})$$
where $\{x\}$ denotes the ...
0
votes
1answer
200 views
Transitive group actions and homogeneous spaces
Given a topological group $G$ and a space $X$ with a transitive $G$ action, let $G_x$ be the isotropy group of a point. In Folland "A course in harmonic analysis", there is a statement that $X$ is ...
5
votes
1answer
554 views
Lyapunov's Inequality for Weak-Lp Spaces
Let $(X,\mu)$ be a measure space. Suppose that $0 < p_{0} < p < p_{1} < \infty$ and $\frac{1}{p} = \frac{1-\theta}{p_{0}} + \frac{\theta}{p_{1}}$ for some $\theta \in (0,1)$. If $f \in ...
3
votes
1answer
90 views
Fourier analysis on groups and paths in a Cayley graph
If one takes a cyclic group and a function on this group, and performs harmonic analysis on it (classical Fourier analysis), the result is a set of coefficients, each one of them corresponding to ...
7
votes
1answer
172 views
Fourier transform of function in $L^{4/3}$
Suppose $f \in L^{4/3}(\mathbb{R}^2)$ and denote its Fourier transform by $\mathscr{F}(f)$. Is it true that the function $g:\mathbb{R}^2 \rightarrow \mathbb{C}$ defined by
...
5
votes
1answer
123 views
In what locally compact abelian groups does $\mathbb{Q}$ embed densely?
I know that there is classification of local fields, but here is a closely related question: Can the additive group of $\mathbb{Q}$ be a proper dense subgroup of a locally compact abelian group, whose ...
4
votes
1answer
2k views
Criteria for swapping integration and summation order
I have a function (a potential from an electrostatic potential via a Fourier series) in the form of
$$V(x, y, z)=\sum_n\sum_m \ a(x, n, m) b(y, n) c(z, m) \int\int f(u, v) d(u,n) e(v,m)
du\, dv$$
...
2
votes
1answer
115 views
Restricted Direct Products in Koch's Number Theory
On p.353 of Number Theory: Algebraic Numbers and Functions by Helmut Koch, he considers a group $G$ which is the restricted direct product of the locally compact abelian groups $G_i$ with respect to ...
2
votes
2answers
105 views
regularity of $d\mu=u dx$
Let $G$ be a abelian, locally compact group with a Haar measure $dx$ on $G$. We know that every Haar measure is inner regular, ie... for any open subset $U$ of $G$, then ...
1
vote
1answer
173 views
Hardy-Littlewood maximal function of a Lipschitz function
In a book, it is said that Hardy-Littlewood maximal function of a Lipschitz function is also Lipschitz. How do we prove this?
+) For Hardy-Littlewood maximal function, see: ...
1
vote
0answers
156 views
Harmonic measure
could anybody will help me to do this problems:
Let $\mathcal D$ be the unit disk a Set $E\subseteq\partial\mathcal D$ has harmonic measure identically $0$ with respect to $\mathcal D$. What can you ...
1
vote
1answer
369 views
Is this a square wave signal?
i have a decomposition of a square wave signal:
$$ y = \frac{4h}{\pi}(\sin(x) + \frac{1}{3}\sin(3x) + \frac{1}{5}\sin(5x) + ...) $$
I computed the fundamental wave and 2 harmonic waves:
$$ U_{r0} = ...
2
votes
1answer
76 views
Convolution on noncommutative group algebras
If $G$ is a non-Abelian locally compact group, and $f$ is in $L^1{(G)}$ and $u$ is in $L^{\infty}(G)$, and $f\ast u=0$ can it be concluded that $u\ast f=0$?
3
votes
0answers
134 views
Extending a convolution operator from $L^p(\mathbb{R}^d)$ to $L^p(\mathbb{R}^d;L^q(\Omega))$
Let $1<p,q<\infty$ and $\Omega$ some $\sigma$-finite measure space. Let $T$ denote a bounded convolution operator on $L^p(\mathbb{R}^d)$ with scalar valued kernel $K$ which is locally integrable ...
1
vote
0answers
121 views
Estimate the Hilbert transform
Let $1\leq p<∞$: Suppose that there exists a constant $C>0$ such that for all $f\in S(\mathbb{R})$ with $L^p$ norm one we have $$\biggl|\{x:|H(f)(x)|>1\}\biggr|\leq C.$$ Here $H(f)$ is ...
0
votes
2answers
508 views
Sine wave harmonics, sawtooth waveform modified
I am trying to find the formula to generate the waveform below. By using harmonics on standard sine waves and then combining the outcomes, I have managed to generate, triangle, sawtooth and square ...
10
votes
1answer
195 views
Suppose $\phi$ is a weak solution of $\Delta \phi = f \in \mathcal{H}^1$. Then $\phi\in W^{2,1}$
I'm trying to prove the statement in the title in as simple a way as possible. It is Theorem 3.2.9 in Helein's book "Harmonic maps, conservation laws, and moving frames", although it is not proved ...
3
votes
2answers
318 views
Why are translation invariant operators on $L^2$ multiplier operators
For $m \in L^\infty$, we can define the multiplier operator $T_m \in L(L^2,L^2)$ implicitly by
$\mathcal F (T_m f)(\xi) = m(\xi) \cdot (\mathcal F T_m)(\xi)$
where $\mathcal F$ is the Fourier ...
2
votes
1answer
246 views
Convolution on group with measure
I was wondering about the generalization of the concept of convolution from the familiar one on real spaces and how many properties still remain.
For convolution on Lebesgue-integrable real-valued ...
2
votes
1answer
165 views
Convolution inequality
Let $u$ and $v$ be two $L^1(\mathbb{R})$ functions such that
$\|u\|_{L^1} \le \|v\|_{L^1}$ and $f$ is non-negative $L^1(\mathbb{R})$
with non-negative inverse Fourier transform.
Is it true that for ...
1
vote
2answers
77 views
Sequence of smooth functions whose image under a maximal operator diverges in $L_p$ norm
For functions $f: \mathbb{R} \rightarrow \mathbb{C}$, define
$$
M f(x) = \sup_{t >0} \frac{1}{2}| f(x+t) + f(x-t) |.
$$
Given $p \geq 1$, I want to construct a sequence of smooth functions ...
4
votes
1answer
82 views
Bound on the relative measure of $\delta$-neighbourhoods of compact sets using the Hardy-Littlewood maximal theorem
Notation: $|A|$ is the Lebesgue measure of $A \subset \mathbb{R}^d$, and $A_\delta = \{ x : \text{dist}(x,A) \leq \delta \} $ is the $\delta$-neighborhood of $A$.
I want to show that there is a ...
1
vote
0answers
88 views
Intuition for Calderon-Zygmund operator?
What is the best intuition for Calderon-Zygmund operators? Why are they so important in singular integrals, and complementary, which singular integrals don't they cover?
11
votes
1answer
427 views
What are the differences and relations of Haar integrals, Lebesgue integrals, Riemann integrals?
Are Riemann integrals special cases of Haar integrals? Why do we need the invariant property under some actions of groups in the definition of Haar integrals? For example, if we have a group of real ...
11
votes
3answers
428 views
learning algebra and harmonic analysis
I've revised my question a bit in response to the (very helpful) advice so far--
I have an engineering background but am interested in learning abstract harmonic analysis. My interest is rather ...
4
votes
2answers
195 views
Operators commuting with translations
Let $T$ be a bounded linear operator on $L^2(\mathbb R)$. So, let us now assume that $T$ commutes with the translations $\tau_x$. How do I now show that $T$ is given by a convolution with respect to a ...
4
votes
3answers
266 views
Derivatives distribution
Let $f$ be a distribution on $\mathbf{R}^n$ (in the Schwartz sense) such that
$$\frac{\partial f}{\partial x_i} = 0 \text{ for $i = 1, \ldots, n$.}$$
Then how to prove that $f$ is a constant? I had ...
3
votes
1answer
273 views
Fourier transform of a special Schwartz function
In Classical Fourier Analysis by Loukas Grafakos we have in Proposition 2.3.25 the following definition for $\mathcal{S}_\infty(\mathbf{R}^n)$, namely that these are all the Schwartz functions $\phi$ ...
