1
vote
2answers
144 views

Measures, orthogonal to holomorphic functions

Let $G$ be a domain in $\mathbb{C}^{d}$ and let $H\left(G\right)$ be the space of all holomorphic functions on $G$. My question is how to characterize all such (Radon) measures $\mu$ on $G$, that ...
1
vote
1answer
165 views

System of quadratic complex equations

I want to solve this system of N non-linear equations without using a numerical method: $x_{k}^{2}= \alpha_{k }+ \sum\limits_{m=1}^{N} (\beta_{km} x_{m} + \psi_{km} x_{m}^{*})$ With $\left| ...
1
vote
1answer
296 views

Infinite product's question

Given a pair of strictly increasing functions $f,g:\mathbb{N}\to \mathbb{N}$ define: $P_N(f,g)\doteq \left(z\in \mathbb{C}\mapsto \prod_{i=1}^{f(N)}\left(1+\frac{z}{v_i(N)}\right)\in ...
19
votes
2answers
506 views

Which smooth compactly supported functions are convolutions?

If $f,g$ are smooth functions with support in the interval $[-r,r]$ for some $r>0$, then their convolution $f*g$ is smooth with support in $[-2r,2r]$. My question is about the converse: Given ...
0
votes
1answer
159 views

The dual space of the Dirac measures on an Abelian group

Let $G$ be a Hausdorff locally-compact Abelian group and $L^2(G)$ the Hilbert space of two-integrable complex functions on the group. Question. What would be natural vector space $\mathcal{R}$ ...
1
vote
2answers
171 views

Analytic continuation for PI(1+z^(4^n))

How to do analytic continuation for following function? $$f(z) = \prod_{n=0}^{+\infty} {(1+z^{4^n})}$$ Evidently it satisfies $f(z)f(z^2)=\dfrac{1}{1-z}$...
1
vote
0answers
243 views

complex contour integral calculation after Möbius transformation

Good day to everyone. In my scientific research I've got stuck with a contour integration problem. I would like to evaluate the following integral: $$I=\int_0^{\infty } \frac{e^{\frac{\alpha -\mathrm ...
0
votes
1answer
328 views

Question on Hartogs's Extension Theorem

Does Hartogs's extension theorem hold if one replaces the word holomorphic by analytic (of course still in several variables)? For Hartogs's Extension Theorem see here: ...
0
votes
2answers
339 views

Polynomial growth of Fourier transforms

I am looking for a theorem that guarantees the polynomial growth of a function $f$ defined by a Fourier integral, that is, when $$f(x)=\int_{-\infty}^{\infty}F(y)e^{ixy}dy.$$ I am only interested in ...
2
votes
4answers
395 views

Nth root of a matrix as an analytic function?

Let $A$ be a $k \times k$ invertible matrix over complex numbers. If it possible to write its nth root as an analytic function (i.e. power series in $A$)? EDIT: Complex coefficients can be functions ...
0
votes
1answer
228 views

Blaschke condition on upper half plane

if f is in $H^{1}$ the Hardy space with integrable norm, and if f is not identically zero, then the zeroes of f (certainly countable in number) satisfy the Blaschke condition.Can any "Blaschke ...
9
votes
2answers
515 views

Complexifying a real Banach space and its dual

A standard way to define the "complexification" $E_\mathbb{C}$ of a real Banach space $E$ is to define a complex linear structure on $E\times E$ by (1) $(x,y)+(u,v)=(x+u, y+v)$, (2) ...
13
votes
2answers
387 views

Spectra of elements of a Banach algebra and the role played by the Hahn-Banach Theorem.

This problem was posed on Math StackExchange some time ago, but it did not garner any solutions there. I think that it is interesting enough to be posed here on Math Overflow, so here it goes. Let $ ...
2
votes
1answer
325 views

Constructing the imaginary part of a holomorphic function

Hallo, Let $f: U \rightarrow \mathbb{R}$ be a analytic function, where $U \subset \mathbb{C}^{n}$ is a open set (paracompact, starshaped or convex i.e. sufficiently nice). Does there exist a function ...
3
votes
2answers
441 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 ...
1
vote
1answer
110 views

About principal values and Wirtinger derivative

Let $K$ be a compact of the plane of Lebesgues measure 0 and $\Omega$ a domain containing $K$. Denote by $E$ the vector space of functions that are holomorphic on $\Omega - K$. I'm interested in ...
0
votes
0answers
156 views

Harmonic Function?

Hi, Let $\varphi : U \rightarrow \mathbb{C}$ be a holomorphic function, where $U \subset \mathbb{C}^{n}$ containing $0$. Is the function $u(x_{1}, ..., x_{n}, y_{1}, ..., y_{n}) := Im( ...
-2
votes
1answer
321 views

holomorphic extension of a function [closed]

hi, I have the following question: let $U \subset \mathbb{C}^{n}$ be some open set containing zero. let $\tilde{U} = U \cap \mathbb{R}^{n}$. assume we have a real-valued analytic function $f : ...
1
vote
1answer
306 views

Almost analytic continuation

Let $f\in S^{\alpha}$ for some $\alpha \in \mathbb{R}$(which means that f is smooth and satisfies $|D^{\beta}f|\leq C(1+|x|)^{\alpha-\beta}$),a function $\tilde{f}$ on $\mathbb{C}$ is called an almost ...
0
votes
1answer
862 views

In Fourier Transforms: Positive Definite Functions, Bochner's Theorem, and Derivatives

I've been reading about Bochner's Theorem lately, but when I apply it to the derivative of a function, I seem to get a contradiction with the theorem. "Bochner's theorem states that a positive ...
0
votes
1answer
312 views

An asymptotic series for the digamma function

As we know, there is an asymptotic series for the digamma function when $z>0$ is a real number. $$ \psi(z)=\ln z+\sum_{n=1}^{\infty}{\frac{B_n}{nz^n}} $$ $B_n$ is the first Bernoulli numbers. How ...
4
votes
0answers
275 views

Laplace Transform: Are there theorems similar to the Bernstein Theorem?

Bernstein's Theorem states, that if a function is completely monotonic, then it is the Laplace transform of an $L^1$-function. (E.g. Widder, "The Laplace Transform", Chapter IV, Theorem 19b) Are ...
1
vote
2answers
330 views

Reversed disc algebra?

Take $U=\mathbb{D}_2\setminus \overline{\mathbb{D}_1}$ ($\mathbb{D}_r$ is the open disc centered at 0 with radius $r$) and consider the space $A(U)$ of all functions on $\overline{U}$ which are ...
11
votes
3answers
1k views

What holomorphic functions are limits of polynomials?

Let $\Omega$ be a connected open set in the complex plane. What is the closure of the polynomials in $\mathcal{H}(\Omega)$ the set of holomorphic functions on $\Omega$? The topology is the usual ...
3
votes
1answer
392 views

Factorization in the Wiener algebra on the unit disc.

Consider the Banach algebra $W^+=\ell^1(\mathbb{Z}^+)$, viewed upon as the analytic functions $f$ on the unit disc $\mathbb{D}$ such that $$\|f\|=\sum_{k\ge0}|a_k|<\infty$$ where $$f(z)=\sum ...
2
votes
2answers
482 views

L^2 space of holomorphic functions with given weight

Hi folks, what is known about the $L^2$ space of holomorphic functions of 1 complex variable with the scalar product $\langle f, g \rangle = \int dzd{\bar z} \frac{ {\bar f(z)} g(z) }{(1 + z{\bar ...
20
votes
3answers
906 views

Universality of zeta- and L-functions

Voronin´s Universality Theorem (for the Riemann zeta-Function) according to Wikipedia: Let $U$ be a compact subset of the "critical half-strip" $\{s\in\mathbb{C}:\frac{1}{2}<Re(s)<1\}$ with ...