1
vote
0answers
39 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). ...
0
votes
0answers
24 views

A question about the convention for the Plancherel measure on $\mathbb{H}^n$

Say I have to calculate the quantity, $Log Tr [ -\Delta - \frac{1}{4} + m^2]$ on $H^n$. Then looking up the spectral measure $\mu(\lambda)$ and the eigenvalues of the Laplacian ($= -\Delta = - ...
2
votes
2answers
229 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 $M$, ...
2
votes
1answer
152 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 ...
1
vote
0answers
99 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 ...
2
votes
2answers
251 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) = ...
1
vote
2answers
301 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 ...
2
votes
1answer
322 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 $B(x)$ ...
0
votes
0answers
203 views

is there a relation between the complex Hardy spaces and the Hardy spaces of harmonic analysis?

Maybe my question is just a matter of knowing the right equivalent definition. The question is whether there is some relation between $ H^p(D^2)$, defined as the space made of the analytic ...
2
votes
6answers
851 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 ...
2
votes
1answer
1k 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 ...