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, ...
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 ...
11
votes
1answer
328 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 ...
2
votes
1answer
184 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 ...
6
votes
1answer
289 views
A net version of dominated convergence?
Let $G$ be a locally compact Hausdorff Abelian topological group. Let $\mu$ be a Haar measure on $G$, i.e. a regular translation invariant measure. Let $f$ be fixed in $\mathcal{L}^1(G, \mu)$. ...
14
votes
4answers
364 views
About a harmonic series problem : How to prove that $\sum_{n=1}^{\infty}\frac{H_n}{n^3}=\frac{\pi^4}{72}$
How to prove that
$$\sum_{n=1}^{\infty}\frac{H_n}{n^3}=\frac{\pi^4}{72}$$
$H_n$ is the n th harmonic number
10
votes
3answers
471 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, ...
82
votes
4answers
2k views
What do modern-day analysts actually do?
In an abstract algebra class, one learns about groups, rings, and fields, and (perhaps naively) conceives of a modern-day algebraist as someone who studies these sorts of structures. One learns about ...
11
votes
2answers
242 views
A series involves harmonic number
How do we get a closed form for
$$\sum_{n=1}^\infty\frac{H_n}{(2n+1)^2}$$
4
votes
1answer
375 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 ...
13
votes
3answers
355 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 ...
6
votes
1answer
183 views
$1/|x|^n$ is not integrable
Let $\mu $ be a positive Borel measure on $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{d}$ such that $\mu \left( B\left( a,r\right) \right) \leq Cr^{n}$ for some
$n\in (0,d]$ ...
3
votes
2answers
320 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 ...
4
votes
3answers
268 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 ...
2
votes
1answer
139 views
Prove partial derivatives of uniformly convergent harmonic functions converge to the partial derivative of the limit of the sequence.
I think the title says it all.
If you have a sequence of harmonic functions from a bounded complex domain to the real numbers, show that on a subset at a positive distance from the boundary of the ...
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.
0
votes
1answer
201 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 ...
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 ...
5
votes
1answer
126 views
A Schwartz function problem
Let f be a strictly positive Schwartz function on $\mathbb R$. Does it imply $\sqrt f$ is a Schwartz function on $\mathbb R$?
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
$$V(x, y, z)=\sum_n\sum_m \ a(x, n, m) b(y, n) c(z, m) \int\int f(u, v) d(u,n) e(v,m)
du\, dv$$
...
3
votes
1answer
32 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
2answers
208 views
Homogeneous function in bounded mean oscillation BMO($\mathbb R^n$) space
Let me recall some notations: The mean oscillation of a locally integrable function $u$ (i.e. a function belonging to $ L^1_{\textrm{loc}}(\mathbb{R}^n))$ over a cube Q in $\mathbb R^n$ (which has ...
2
votes
2answers
282 views
amplitude of sine wave with multiple frequencies
I'm having some troubles determining the amplitude/magnitude of the following equation.
$$
A\cos(2\omega t+\beta_1)+B\cos(3\omega t+\beta_2)+C\cos(5\omega t+\beta_3)
$$
Since each part is at a ...
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 $\varphi$ ...
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 ...
1
vote
1answer
24 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 ...
1
vote
1answer
79 views
Dirichlet problem: Is the Poisson Integral always a solution?
Let $f$ be continuous on the sufficiently smooth boundary $\partial D$ of a domain $D \subset \Bbb R^n$.
Is the Poisson integral of $f$,
$$
Pf(x)=\int_{\partial D} f(t) ...
1
vote
0answers
119 views
Find a sequence of function in the Schwartz space $S(\mathbb R)$ which does not converge in $S(\mathbb R)$
Show there exists a sequence $\{f_n\}$ in the Schwartz space $S(\mathbb R)$ with limit $f$ for which
$$
\lim \|f_n\|_{u,v} \text{ induced that } f \not\in S(\mathbb R) \text{ for some } u,v.
$$
But
...
