All Questions
Tagged with cv.complex-variables harmonic-analysis
24 questions with no upvoted or accepted answers
10
votes
0
answers
207
views
Projective tensor squares of uniform algebras
In discussion with a colleague recently (Jan 2017),
$\newcommand{\AD}{A({\bf D})}\newcommand{\CT}{C({\bf T})}$
I was reminded that if $A(D)$ denotes the disc algebra and $\iota: \AD\to \CT$ is the ...
- 25.8k
8
votes
0
answers
339
views
The Cauchy Transform, and the convergence of the Fourier-Stieltjes transforms of a sequence of measures
Let $C\left(\mathbb{R}/\mathbb{Z}\right)$ denote the Banach space of continuous, $1$-periodic complex-valued functions on the unit interval, let $M\left(\mathbb{R}/\mathbb{Z}\right)$ denote its dual, ...
- 1,284
7
votes
0
answers
168
views
The relation between Wolf's and Teichmüller's parametrization of the Teichmüller space
Let $\mathcal{T}_g$ be the Teichmüller space of Riemannian surface structures on an oriented 2-dimensional manifold of genus $g$. Fix a point $S \in \mathcal{T}_g$. There are two different ways to ...
- 1,170
6
votes
0
answers
219
views
Extremal polynomial majorants of $\log{|f|}$: a multivariate extension of a theorem of Carneiro and Vaaler
Carneiro and Vaaler have proved, as an application of their work on Beurling-Selberg extremal majorants, that for any non-zero complex polynomial $f(z) \in \mathbb{C}[z]$, the infimum value of the ...
- 13.8k
4
votes
0
answers
173
views
On the best constant for Carleson's embedding theorem
In "Interpolations by bounded analytic functions and the corona problem", Carleson introduced Carleson measures (for Hardy spaces) and proved the famous embedding theorem according to which $...
- 81
4
votes
0
answers
104
views
Walsh-Lebesgue type theorem in $\Bbb R^{2m}$ for $m>1$
Is someone aware of any analogue of the Walsh-Lebesgue theorem in $\mathbb{R}^{2m}$ for $m>1$ and dealing with polyharmonic polynomials?
In this post, $\phi$ is said to be polyharmonic in $\mathbb{...
- 41
4
votes
0
answers
210
views
Inclusion of Hardy spaces
It is well-known that any convergence in $L^p$ for $p \in [1,\infty]$ implies convergence in $L^1_{\text{loc}}$ by Hölder's inequality.
It is also known that for $p>1$ it holds that $L^p(\mathbb R)...
3
votes
0
answers
167
views
Does the maximum principle hold in this pluriharmonic setting?
Let $U \subseteq \mathbb{C}^m$ be open, and let $F: U \to \mathbb{C}$ be a holomorphic function, with real part $u$. We are given a subset $S \subseteq U$ given by finitely many real equalities and ...
- 5,215
3
votes
0
answers
250
views
Dimension of roots of irreducible Schur polynomial on unit circle
Let $s_\lambda(x_1,\ldots,x_n)$ be a Schur polynomial in $\mathbb{C}[x_1,\ldots,x_n]$ with $\lambda=(\lambda_1\geq\lambda_2\geq\ldots\geq\lambda_n=0)$ and $\gcd(\lambda_1+n-1,\lambda_2+n-2,\dots,\...
- 31
2
votes
0
answers
120
views
Does a holomorphic function with logarithmic growth at the boundary have $L^2$ boundary values?
Let $f(z)$ be a holomorphic function on the unit disc, with logarithmic growth at the boundary:
$$
|f(z)| = \mathcal O\bigg(\log\Big(\frac{1}{1-|z|}\Big)\bigg).
$$
Does it follow that the (...
- 43.2k
2
votes
0
answers
88
views
Link between subharmonic and subanalytic functions
Consider $\Omega$ an open set of $\mathbb{C}$ and $f : \Omega \to \mathbb{R}$ a $C_{\infty}$-function. The following two definitions are well-known (at least the first one) but I prefer to recall them ...
- 1,017
2
votes
0
answers
136
views
To find a positive function with compact spectrum
Let
$e_1=(0,1)^T$,
$$
S=\left\{x\in \mathbb{R}^2\Big| \frac{|\langle x, e_1\rangle|}{|x|}>\delta>0\right\},
$$
is a cone in $\mathbb{R}^2$.
I want to find a non-trivial smooth function ...
- 29
1
vote
0
answers
338
views
Recognizing when a $2\pi$-periodic function is a shifted sine
Consider a real analytic function $f(x)$ with a period of $2\pi.$ Let $t\in \operatorname{Range}(f(x))$, $$I(t)=\int_{0}^{2\pi}\log |t-f(x)|dx.$$ Prove that $I(t)$ is equal to a constant if and only ...
- 21
1
vote
0
answers
51
views
Application of the $\operatorname{BMO}$, $H^1$ duality
Let $f\in \operatorname{BMO}(\partial \Delta)$, then there exists a Carleson measure $\mu$ in $\Delta$ such that
$$f(\zeta)-\int_{\Delta}P_{z}(\zeta)d\mu(z)\in L^{\infty}(\partial \Delta),\ \zeta\in\...
- 111
1
vote
0
answers
47
views
Holomorphic "quasi-interpolation" of a function sequence
I am interested in some sort of analytic interpolation. A toy version of my problem is as follows. Let $V \subset \mathbb{C}$ be a complex neighborhood of $[0,1]$. Assume there is some bounded ...
- 981
1
vote
0
answers
88
views
Invertibility of Hankel operators?
Let $D$ be the unit disc in the complex plane and $P$ the Bergman projection mapping $L^2(D)$ onto the closed subspace $A^2(D)$ of holomorphic square-integrable functions (w.r.t. Lebesgue measure). ...
- 2,479
1
vote
0
answers
130
views
Question about a oscillatory integrals on manifold
Let $M$ be a compact oriented Riemannian manifold without boundary.
Set $f(x)=a(x)+\sqrt{-1}b(x)$ be a complex-valued function on $M$,
where $a(x),b(x)$ are real-valued function on $M$.
Then, how to ...
- 381
0
votes
0
answers
57
views
Double-periodic functions with (possible) poles
Consider the set of double-periodic function $f:\mathbb C/(\mathbb Z+i \mathbb Z) \setminus \{z_0\} \to \mathbb C$, where $z_0$ is a fixed point inside $\mathbb C/(\mathbb Z+i \mathbb Z),$ that have a ...
0
votes
0
answers
94
views
The asymptotic behaviour of the Fourier transform of a certain class of radially symmetric functions
Fix $\theta\in (-\pi/2,\pi/2)$ and let $a>0$. Suppose that $f:\mathbb{C}\rightarrow \mathbb{C}$ is analytic in $S:=\{z\in \mathbb{C}: |\arg{z}|<\pi/2\}$ and
$$|f(z)|\sim |z|^{-a},\qquad |z|\to \...
- 852
0
votes
0
answers
173
views
Function Spaces on the Open Unit Disk defined by Hardy Space norms
I've been reading up on Hardy spaces and (sub)harmonic functions over the open unit disk $\mathbb{D}\subset\mathbb{C}$, and I've found myself working with atypical objects in mostly-typical situations....
- 1,284
0
votes
0
answers
47
views
Oscillatory integral independent of a parameter
Let $h: \mathbb{R} \mapsto \mathbb{C}$ be a positive definite function, continuous at the origin. (In fact, $h$ is the Fourier transform of a finite measure). Define the oscillatory integral
$$Q(t) :=...
- 222
0
votes
0
answers
116
views
Dimension of the set of the polynomial growth harmonic function on the hyperbolic plane
We consider the hyperbolic plane and the harmonic function there. Pick any point $p$. Let $H_n, n \in\mathbb N$ be the set of the harmonic functions $f$ such that $|f(x)|\leq c(1+ d(x,p))^n$.
What is ...
- 5,043
0
votes
0
answers
49
views
Non interacting complex unit
How to work with two non interacting complex units say i and j. These two imaginary complex unit represent different quantities. For example i is for periodicity in theta and j is for frequency or ...
- 1
0
votes
0
answers
256
views
Explicit formula for Bergman kernel on the unit ball
On page 173 in Krantz's book "Explorations in Harmonic analysis" in the proof of Lemma 7.1.21 there is a part that I really don't understand. What I don't understand is why is $$\sum_{\alpha}\frac{z^{\...
- 325