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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 ...
user35486's user avatar
  • 141
2 votes
1 answer
850 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 \right|...
1 vote
0 answers
246 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 ...
Rick Farr's user avatar
23 votes
2 answers
2k 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 ...
Gandalf Lechner's user avatar
0 votes
1 answer
715 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 $\mathcal{R}$ of ...
Juan Bermejo Vega's user avatar
4 votes
0 answers
112 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.
Koushik's user avatar
  • 2,106
4 votes
1 answer
1k 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 ...
Koushik's user avatar
  • 2,106
2 votes
2 answers
212 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 $f(z)f(z^2)=\dfrac{1}{1-z}$...
Lwins's user avatar
  • 1,551
2 votes
1 answer
265 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 $ f(z)=\...
Koushik's user avatar
  • 2,106
-1 votes
1 answer
211 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 ...
yaoxiao's user avatar
  • 1,706
2 votes
0 answers
605 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 ...
Caran-d'Ache's user avatar
3 votes
1 answer
280 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) ...
Loïc Teyssier's user avatar
2 votes
0 answers
122 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 forms....
user36504's user avatar
0 votes
2 answers
1k 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: http://en.wikipedia.org/wiki/...
bernard's user avatar
  • 53
0 votes
2 answers
713 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 ...
Kevin Smith's user avatar
  • 2,480
2 votes
4 answers
1k 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 ...
Piotr Migdal's user avatar
  • 1,612
2 votes
1 answer
804 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 ...
Koushik's user avatar
  • 2,106
11 votes
2 answers
2k 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) $(a+ib)(x,y)=(ax-...
Fred Dashiell's user avatar
1 vote
0 answers
237 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 $|...
Frank's user avatar
  • 9
18 votes
3 answers
1k 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 $ \...
Leonard's user avatar
  • 816
3 votes
1 answer
1k 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 ...
hapchiu's user avatar
  • 339
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 ...
hapchiu's user avatar
  • 339
1 vote
1 answer
215 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 ...
Albert's user avatar
  • 377
1 vote
2 answers
263 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.
Makoto Kato's user avatar
  • 1,169
-2 votes
1 answer
1k 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 : \tilde{...
bruno's user avatar
  • 29
1 vote
2 answers
437 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 ...
Stephan Wehner's user avatar
1 vote
1 answer
1k 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 ...
user23078's user avatar
  • 1,644
3 votes
1 answer
226 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 ...
Emilio Pisanty's user avatar
1 vote
1 answer
3k 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 ...
Robert's user avatar
  • 11
1 vote
1 answer
779 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 ...
Lwins's user avatar
  • 1,551
4 votes
3 answers
464 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 ...
Euplio M.'s user avatar
2 votes
0 answers
262 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 ...
Joshua Isralowitz's user avatar
4 votes
0 answers
500 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 ...
florian's user avatar
  • 93
2 votes
2 answers
386 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 ...
JSanderson's user avatar
5 votes
1 answer
510 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 1-\tfrac{...
RogersFR's user avatar
0 votes
1 answer
2k 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, ...
john mangual's user avatar
  • 22.8k
1 vote
1 answer
312 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} ...
Joshua Isralowitz's user avatar
15 votes
3 answers
4k 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 ...
Olivier Bégassat's user avatar
12 votes
1 answer
735 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....
Zen Harper's user avatar
  • 1,990
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 ...
Hadi's user avatar
  • 741
7 votes
3 answers
1k 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. Here are my questions, and I'd be grateful of any reference or information. Let $\...
Pietro Majer's user avatar
  • 60.5k
4 votes
1 answer
645 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 a_kz^k$...
AD - Stop Putin -'s user avatar
38 votes
2 answers
13k views

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

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] Louis ...
Zen Harper's user avatar
  • 1,990
4 votes
1 answer
6k 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 $w$ ...
Mermoz's user avatar
  • 167
18 votes
1 answer
3k 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 ...
Zen Harper's user avatar
  • 1,990
2 votes
2 answers
679 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 z})^...
Daniel's user avatar
  • 362
26 votes
3 answers
2k 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 ...
M.G.'s user avatar
  • 7,127

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