1
vote
1answer
166 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
298 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 ...
2
votes
1answer
186 views

ideals in the disk algebra

Let A be the disk algebra, of continuous functions on the closed disk holomorphic on the interior, with sup-norm denoted || . || . Let x be an interior point of the disk. Does there exist a ...
2
votes
1answer
530 views

Absolute convergence of logarithm of polynomial with positive coefficient ($\ln G(z) = \sum\limits_{i = 0}^\infty {{q_i}{z^i}} $)

Special problem: Let $G(z)$ be a probability generating function(pgf, the $z$ can be seen as real number or complex number), that is $$G(z) = \sum\limits_{i = 0}^\infty {{p_i}{z^i}} ,(\left| z ...
0
votes
0answers
65 views

Finding the Fractional Derivative of This Function [duplicate]

I've been trying to find an answer to this question, and it seems as though the question has gone unanswered. The question regards the derivative of $f(x)=1+n^{-x}$ where $n$ is a natural number. Is ...
0
votes
0answers
128 views

Fractional Derivative of A specific function

I've tried and tried to do this problem myself, but I've hit some snags on the way. I'm trying to take the fractional derivative of: $f(x)=1+n^{-x}$ where n is an integer and $n\geq2$ and $x>1$. ...
0
votes
0answers
136 views

Fractional Derivatives Of Sums

I have a question regarding the definition of a fractional derivative. I've searched, but I can't find a definition of fractional derivatives that explain the concept in terms of an operator on some ...
19
votes
2answers
511 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 ...
4
votes
0answers
76 views

status of Invariant subspace problem on Krein Space

What is the status of Invariant subspace problem on Krein Space? What sort of developments have taken place in this area.
5
votes
1answer
224 views

definition of accretive operator

A relation T with domain and range in a Hilbert space is said to be accretive if the transformation $ (T − \lambda)/(T + \bar \lambda\ ) $ with domain and range in the Hilbert space is contractive for ...
2
votes
1answer
215 views

hayman's result for $ A^2(D) $

Consider injective homolomorphic functions $f:\mathbb D\to \mathbb C$ on the unit disk $|z|\leq 1$, normalized by the conditions $f(0)=0$ and $f'(0)=1$. Thus for $|z|\leq 1$ we have $ ...
-1
votes
1answer
138 views

Stone Cech compactification for exponential map

Recently I met with a problem related to Stone-Cech Compactification theorem in Furstenberg's famous paper "non-commuting product." I try my best to understand Stone-Cech compactification theorem by ...
1
vote
0answers
250 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 ...
1
vote
1answer
200 views

Composite families of formal power series over $\mathbb C$ as algebraic variety

I was led to prove that the set of composite families $(f_j)_{j \leq k}$ of germs at $0\in \mathbb C^m$ of a holomorphic function (composite = sharing a common divisor belonging to the maximal ideal) ...
2
votes
0answers
89 views

A parametrix for the $\bar\partial$ operator adapted to a holomorphic foliation

Let $X$ be a compact complex manifold with (regular) holomorphic foliation given by a holomorphic subbundle $\mathcal F$ of the tangent bundle. The foliation induces a filtration on differential ...
0
votes
1answer
335 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: ...
1
vote
0answers
198 views

bivariate polynomial

Hello, Let $p(x,y) = \sum_{m=1}^M\sum_{n=1}^N a_{m,n}x^{m-1}y^{n-1}$ be a bivariate polynomial where $\{a_{m,n}\}$ are complex. If $(x_k, y_k), k=1,2,\cdots, MN-1$ are roots of $p(x,y)=0$ where ...
2
votes
1answer
340 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 ...
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( ...
0
votes
2answers
160 views

Books on real and/or complex analytic functions on Banach spaces taking values in Banach spaces

I'm looking for good textbooks on the subjects. If you know one(s), please let me know.
1
vote
2answers
378 views

Analytical predicate for integers over complex numbers

A complex number $z$ is an integer if and only if $\sin(\pi z)=0$. It follows that a complex number $z$ is an integer if and only $\sin^2(\pi z) = 0$. So for a real analytic function $f$ and any real ...
3
votes
1answer
140 views

Analytic continuation of instantaneous eigenstates of a time-dependent hamiltonian

We are considering the instantaneous eigenstates of an analytically time-dependent hamiltonian and I would like to know how legitimate it is to extend them to the complex plane. Specifically, our ...
4
votes
3answers
362 views

What classes of functions are closed under all rescalings?

Let us denote by the symbol $\mathcal{G}$, a group of functions $f: \mathbb{R} \rightarrow \mathbb{R}$ (with the composition operation) that is additionally closed under all affine change of variables ...
0
votes
0answers
205 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
0answers
185 views

A specific projection and compactness on the Bargmann-Fock space

Let $F_2$ be the Bargmann Fock space defined as the space of entire functions $f$ on $\mathbb{C}$ such that \begin{align*} \int_{\mathbb{C}} |f(z)|^2 e^{- |z|^2} dA(z) \end{align*} ($dA$ is just ...
5
votes
1answer
421 views

The space $H(D)$ of holomorphic functions.

A very natural example of a nuclear Montel space is the space $H(D)$ of all holomorphic functions on the open disc topologized by the family of seminorms $$p_n(f)=\sup\{|f(z)|\colon |z|\leq ...
0
votes
1answer
779 views

What is the orthonormal basis for the Bergman space on the disk?

[EDIT by YC: the original question's title asked about a basis for the Hardy space on the disk. It is clear from the actual question that what was meant was the Bergman space.] In arXiv:0310.5297, ...
1
vote
1answer
235 views

Invertibility of frame/sampling operator on Bargmann-Fock spaces

Let $F_\alpha ^p (\mathbb{C}^n)$ for $1 < p < \infty$ and $\alpha > 0$ be the Bargmann-Fock space defined as the Banach space of entire functions $f$ such that $f(\cdot) e^{- \frac{\alpha}{2} ...
8
votes
0answers
366 views

Parametrisations for Null Temperature Functions: nonuniqueness of solutions to the Heat Equation

Disclaimer I expect this is a highly open problem, but maybe I'm wrong and someone has come up with some answers besides those given here. In any case, all information appreciated, thanks! Definition ...
2
votes
6answers
857 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 ...
4
votes
3answers
368 views

Compactness properties of plurisubharmonic functions

I'm quite interested in this topic, but the main text on Several Complex Variables say little of nothing about it. As it happens, you are immediately led to think of conspiracy, when few main texts ...
26
votes
1answer
6k views

What, exactly, has Louis de Branges proved about the Riemann Hypothesis?

Hi, I know this is a dangerous topic which could attract many cranks and nutters, but: According to Wikipedia [and probably his own website, but I have a hard time seeing exactly what he's claiming] ...
4
votes
1answer
2k views

Inverse of a function defined by an integral

Hi, I have a function defined by an integral as follows. $$ z=f(w) = \int_0^w \frac{(\zeta-a_1)^{\alpha_1}(\zeta-a_2)^{\alpha_2}...}{(\zeta-b_1)^{\beta_1}(\zeta-b_2)^{\beta_2}...}\ d\zeta $$ where $w$ ...
12
votes
1answer
1k views

Let a function f have all moments zero. What conditions force f to be identically zero?

Throughout, let $f$ be a Lebesgue measurable function (or continuous if you wish, but this is probably no easier). (Questions with distributions etc. are possible also but I want to keep things simple ...
2
votes
2answers
483 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 ...