All Questions
Tagged with cv.complex-variables harmonic-analysis
73 questions
13
votes
3
answers
710
views
Completeness of nonharmonic Fourier Series
I have the following question:
The Exponential System $(\exp(2\pi i n \cdot ))_{n\in \mathbb{Z}}$ constitutes an orthonormal basis of $L^2([-1/2,1/2])$.
Thus, certainly the oversampled system $\Phi:...
- 131
5
votes
1
answer
342
views
harmonic extension of a curve by different parametrization
Let us consider a curve $\gamma :S^1 \rightarrow \mathbb{R}^3$ (or even a planar convex one if it simplifies). Then I look to the harmonic extension to the disc $h:\mathbb{D}\rightarrow \mathbb{R}^3$ (...
- 914
0
votes
1
answer
289
views
Hilbert transform on boundary value of analytic bounded functions
I am considering the boundary values of a bounded holomorphic functions. Suppose $w$ is a bounded holomorphic function in upper half plane, with continuous and bounded boundary value $f$ on real axis. ...
- 41
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
1
answer
729
views
Reproducing Kernel of a RKHS of continuous functions may not be continuous in two variables together
Let $\mathcal{K}$ be a Hilbert Space of continuous functions on some topological space, where point evaluations are continuous linear functional on $\mathcal{K}$.
That is $\mathcal{K}$ is RKHS, ...
- 3
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
7
votes
1
answer
665
views
Dominated convergence to characteristic function
Let $\phi_m(x):=\chi_{[0,1]} * \chi_{[0,1]} *...* \chi_{[0,1]}$ be the m -times convolution (so $m+1$ characteristic functions are involved).
Then the Fourier transform of this function is given by
$...
- 81
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 $$\sum_{\alpha}\frac{z^{\...
- 325
1
vote
0
answers
88
views
Invertibility of Hankel operators?
Let $D$ be the unit disc in the complex plane and $P$ the Bergman projection mapping $L^2(D)$ onto the closed subspace $A^2(D)$ of holomorphic square-integrable functions (w.r.t. Lebesgue measure). ...
- 2,479
3
votes
2
answers
322
views
The Szego projector, the dual disc bundle $\overline{D}$ and representation of $S^1$ on $H^2$($\overline{D}$)
This construction arises when constructing the Szego projector.
Let's consider the dual disc bundle $\overline{D}$ of a positive Hermitian line bundle ($L$,$h$) over a compact Kahler manifold $M$, i....
- 601
2
votes
1
answer
227
views
Interpolating delta like functions by trigonometric polynomials of bounded modulus and fast decay
Consider a grid of points $T=\{t_0,t_1,\ldots,t_m\}$ with $0\le t_i\le 1$. I would like to find a function $f(t):[0,1]\rightarrow \mathbb{C}$ of the form
\begin{equation*}
f(t)=\sum_{k=-n}^n c_k e^{2\...
- 859
1
vote
0
answers
130
views
Question about a oscillatory integrals on manifold
Let $M$ be a compact oriented Riemannian manifold without boundary.
Set $f(x)=a(x)+\sqrt{-1}b(x)$ be a complex-valued function on $M$,
where $a(x),b(x)$ are real-valued function on $M$.
Then, how to ...
- 381
2
votes
2
answers
313
views
Factorization of antisymmetric bounded holomorphic functions
A basic principle in complex function theory is that one can split off zeros of holomorphic functions in a similar way as for polynomials: If $f$ is holomorphic near $0$ and $f(0) = 0$, then $f(z) = ...
4
votes
1
answer
267
views
variation of the obstacle in the obstacle problem
Suppose $D \subset \Bbb C$ with smooth boundary. Let $f \in C^{1,1}(D)$. Let $\varphi$ be the supremum of all members in the set
$$\lbrace g \in C^{\infty}(\overline{D})| g \ is \ subharmonic \ and \...
- 1,201
1
vote
2
answers
599
views
Harmonic polynomials on complex 2-space
Consider a complex-valued harmonic polynomial $f$ on ${\Bbb C}^2$ and assume that $f(0)=0$. Suppose also that $f$ does not vanish on the unit sphere $S^3\subset{\Bbb C}^2\simeq{\Bbb R}^4$. Does it ...
- 19
0
votes
1
answer
303
views
Fixed norm problem for analytic functions
Hi there,
I have the following problems on my hand:
Given arbitrary positive sub-harmonic function $l(x,y)$ on plane, for all analytic functions: $f=a(x,y)+ib(x,y)$ on complex plane, consider the ...
- 583
3
votes
2
answers
904
views
Corona Theorem in several variables
Hallo,
I have read about the Corona Theorem (see link:http://en.wikipedia.org/wiki/Corona_theorem). From this one ca deduce that: Let $f_{1}, ..., f_{n}$ be holomorphic bounded functions on the unit ...
- 339
6
votes
1
answer
1k
views
A question about the Beurling-Selberg majorant
Beurling's majorant is defined as the unique entire function $B(z)$ such that the Fourier transform of $B(x)$ is compactly supported in the interval $[-1;1]$, $B(x) \geq \text{sgn}(x)$ and $B(x)$ ...
- 63
12
votes
1
answer
2k
views
Hardy spaces: analysis <---> martingales
Let $H^p$ be the Hardy space of analytic functions on the open unit disk $\mathbb{D}$: $f \in H^p$ if $f$ is analytic on $\mathbb{D}$ and $\sup_{r < 1} \int_0^{2\pi} |f(re^{i\theta})|^p d\theta <...
- 943
9
votes
1
answer
943
views
Removable sets for harmonic functions and Hardy spaces of general domains
Let $\Omega$ be a domain of the complex plane. The Hardy space $H^p(\Omega)$ is defined, for $1 \leq p<\infty$, as the class of functions $f$ that are holomorphic on $\Omega$ such that $|f|^p$ has ...
- 2,154
3
votes
6
answers
1k
views
Reference for complex analysis jargon
I am not a (complex) analyst but it seems that some of the questions I am working on are related to the following concepts:
logarithmic capacity
transfinite diameter
Green's function of a compact ...
- 741
2
votes
1
answer
4k
views
Relation between complex analysis and harmonic function theory [closed]
There are some theorems in harmonic function theory that resemble results in complex analysis, like:
Holomorphic functions and complex functions are analytic;
Cauchy's integral formula in complex ...
13
votes
2
answers
862
views
Motivation for BMO
At the moment, I don't have access to the early 1960's paper of John and Nirenberg that (from what I understand) introduced the space BMO (bounded mean oscillation). Why were John and Nirenberg ...
- 261