All Questions
Tagged with cv.complex-variables harmonic-analysis
73 questions
1
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2
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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
1
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1
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85
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Discrete uniqueness sets for the two-sided Laplace transform?
Let $f : \mathbb R_+ \to \mathbb C$ be a measurable and integrable function where $\mathbb R_+ = [0,\infty)$. The Laplace transform of $f$ is given by
$$
Lf(s) = \int_0^\infty f(x)e^{-sx} \, dx.
$$
A ...
- 190
1
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1
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261
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Beltrami equation with harmonic coefficient
I need to find solutions to the Beltrami equation
$$
\frac{\partial w}{\partial\overline{{z}}}=e^{i\varphi(z)}\frac{\partial w}{\partial z}
$$
for $w=w(z,\overline{z})$ and $\varphi(z)$ some given, ...
- 134
1
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2
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144
views
Measurability of the angular limit function
Let $\mathbb T$ be the unit circle and suppose that $f\in L^1(\mathbb T)$ is real-valued. Then its Poisson integral $F=P[f]$
is real-valued, too. Let
$$Osz[f](e^{i\theta}):=\limsup_{z\to e^{i\...
- 687
1
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1
answer
132
views
Plancharel-Pólya inequality for functions of exponential type
If $f(z)$ is an entire function of exponential type $\tau$ and $p$ a positive number such that that $$\int_{-\infty}^{+\infty}|f(x)|^pdx<\infty$$
then it can be proven that $$\int_{-\infty}^{+\...
- 319
1
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1
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167
views
Sampling set: relatively dense and uniformly discrete
The Paley-Wiener space of a domain $\Omega\subset\mathbb{R}^d$ is the set $$PW_\Omega:=\{f\in L^2(\mathbb{R}^d):\text{supp}\widehat{f}\subset\Omega\}.$$
We say that a discrete set $\Lambda\subset\...
- 319
1
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0
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338
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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
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0
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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
$$f(\zeta)-\int_{\Delta}P_{z}(\zeta)d\mu(z)\in L^{\infty}(\partial \Delta),\ \zeta\in\...
- 111
1
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0
answers
47
views
Holomorphic "quasi-interpolation" of a function sequence
I am interested in some sort of analytic interpolation. A toy version of my problem is as follows. Let $V \subset \mathbb{C}$ be a complex neighborhood of $[0,1]$. Assume there is some bounded ...
- 981
1
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0
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88
views
Invertibility of Hankel operators?
Let $D$ be the unit disc in the complex plane and $P$ the Bergman projection mapping $L^2(D)$ onto the closed subspace $A^2(D)$ of holomorphic square-integrable functions (w.r.t. Lebesgue measure). ...
- 2,479
1
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0
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130
views
Question about a oscillatory integrals on manifold
Let $M$ be a compact oriented Riemannian manifold without boundary.
Set $f(x)=a(x)+\sqrt{-1}b(x)$ be a complex-valued function on $M$,
where $a(x),b(x)$ are real-valued function on $M$.
Then, how to ...
- 381
0
votes
1
answer
118
views
Nonstationary phase method for oscillatory integral
I want to approximate an integral of the form $$\int_a^bf(t)e^{ig(t)}dt,$$where $f(t)$ is smooth, $g(t)$ is real-valued and smooth.
The stationary phase method says that if $t_0\in [a,b]$ is such that ...
- 107
0
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1
answer
128
views
On a theorem by Mooney and Khavin on the weak sequential completeness of the predual of $H^\infty(\mathbb{D})$
There is a theorem by Mooney http://msp.org/pjm/1972/43-2/pjm-v43-n2-p.pdf#page=185 and independently proved by Havin which says that the predual of $H^{\infty}(\mathbb{D}),$
$L^{1}/H^{1}_{0}$ is ...
- 175
0
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1
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303
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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
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
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
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
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0
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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
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0
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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
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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
$$Q(t) :=...
- 222
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0
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116
views
Dimension of the set of the polynomial growth harmonic function on the hyperbolic plane
We consider the hyperbolic plane and the harmonic function there. Pick any point $p$. Let $H_n, n \in\mathbb N$ be the set of the harmonic functions $f$ such that $|f(x)|\leq c(1+ d(x,p))^n$.
What is ...
- 5,043
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0
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49
views
Non interacting complex unit
How to work with two non interacting complex units say i and j. These two imaginary complex unit represent different quantities. For example i is for periodicity in theta and j is for frequency or ...
- 1
0
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0
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256
views
Explicit formula for Bergman kernel on the unit ball
On page 173 in Krantz's book "Explorations in Harmonic analysis" in the proof of Lemma 7.1.21 there is a part that I really don't understand. What I don't understand is why is $$\sum_{\alpha}\frac{z^{\...
- 325