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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
33 votes
1 answer
2k views

Stone-Weierstrass theorem for holomorphic functions?

The Stone-Weierstrass theorem has an analog for the algebras of smooth functions, called Naсhbin's theorem: An involutive subalgebra $A$ in the algebra ${\mathcal C}^\infty(M)$ of smooth ...
Sergei Akbarov's user avatar
28 votes
9 answers
5k views

Applications of algebra to analysis

EDIT: I would like to make a list of modern applications of algebra in analysis. By "modern" I will mean developments since the beginning of the 20th century. It is well known that classical linear ...
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
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
21 votes
7 answers
2k views

Identities and inequalities in analysis and probability

Usually, at the heart of a good limit theorem in probability theory is at least one good inequality – because, in applications, a topological neighborhood is usually defined by inequalities. Of course,...
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
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
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
15 votes
1 answer
1k views

Borel-Écalle re-summation and resurgence: criteria and results

This is about the theory of Borel-Écalle re-summation and resurgence, see Refs below. This states that the perturbative series (say of the vacuum expectation value of an operator $\mathcal{O}$ in ...
wonderich's user avatar
  • 10.5k
14 votes
6 answers
6k views

Russian Equivalent of Big Rudin

Is there any Russian-authored textbook on Analysis equivalent to Big Rudin (Real and Complex Analysis)? I like Russian math textbooks a lot. I am looking for Russian textbooks (either in English or ...
Kumar's user avatar
  • 149
13 votes
1 answer
408 views

Does the $\overline{\partial}$ operator have closed image?

Let $X$ be a complex-analytic manifold, not necessarily compact. Does $\overline{\partial} : C^\infty(X) \rightarrow \Omega^{0,1}(X)$ have closed image with respect to the Fréchet topology given by ...
Daniel Bruegmann's user avatar
13 votes
3 answers
710 views

Completeness of nonharmonic Fourier Series

I have the following question: The Exponential System $(\exp(2\pi i n \cdot ))_{n\in \mathbb{Z}}$ constitutes an orthonormal basis of $L^2([-1/2,1/2])$. Thus, certainly the oversampled system $\Phi:...
dime's user avatar
  • 131
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
11 votes
3 answers
1k views

"Simple" integral equation

Let $H(z)$ be a continuous solution of the problem $$ H(z)=\frac1{1-z}\int_z^1 \frac{2\zeta}{1+\zeta} H(\zeta^2)\,d\zeta,\ \ \ z\in[0,1);\ \ \ H(1)=1. $$ Is it true that $H(0)=1-\ln2$? The question ...
AAK's user avatar
  • 283
11 votes
1 answer
602 views

How do analysts think about functions with poles at all roots of unity?

In branches of algebra impinging on the enumeration of partitions, one often encounters formulas like $$\prod_i \left( \frac{1}{1-q^i} \right)^{n_i}$$ for some integers $n_i$. E.g., with $n_i = 1$, ...
Vivek Shende's user avatar
  • 8,723
11 votes
1 answer
413 views

Estimating the growth of the Taylor coefficients given the growth of the function at the boundary

Let $f(z)=\sum a_nz^n$ be a Taylor series that converges for $|z|<1$ and satisfies $$ |f(z)|\le \frac{1}{(1-|z|)^{k}} $$ for some fixed $k>0$. Question: What can I deduce about the growth of the ...
André Henriques's user avatar
11 votes
1 answer
411 views

A density question for the Hilbert transform

Let $\mathscr Hf$ denote the Hilbert transform of a function $f$ defined on the real-line $\mathbb R$. Are the set of functions $$ \{(f+\mathscr Hf)_{|_{(0,1)}}\,:\, f \in C^{\infty}(\mathbb R)\quad \...
Ali's user avatar
  • 4,143
11 votes
1 answer
582 views

An extension of the Carlson's theorem in complex analysis

For the statement of Carlson's theorem please see, https://en.wikipedia.org/wiki/Carlson%27s_theorem. There is an extension of Carlson's theorem that says that the condition that $f$ needs to vanish ...
Ali's user avatar
  • 4,143
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
11 votes
1 answer
428 views

Maximal ideals of the ring $\mathbb C \{T\}$

Consider the Banach $\mathbb C$-algebra $$ \mathbb C \{T\} = \left\lbrace \sum_{i \geq 0} a_i T^i : \sum_{i \geq 0} |a_i| < \infty \right\rbrace $$ With the norm given by $\| \sum a_i T^i\| = \sum |...
Aitor Iribar Lopez's user avatar
11 votes
1 answer
486 views

Resources for divergent / asymptotic series

This series is divergent; therefore, we may be able to do something with it. -- Oliver Heaviside [Edit (1/14/21) from the answer by Count Iblis to a recent MO-Q on math vids: An enthusiastic intro is ...
Tom Copeland's user avatar
  • 10.5k
10 votes
2 answers
739 views

Unconditionally convergent series in some functional spaces

Linked with this question and discussion (Bilinear product of two summable families), I am very interested in counterexamples/results about the following questions (cf the end). First, I recall that a ...
Duchamp Gérard H. E.'s user avatar
10 votes
1 answer
349 views

On a variant of Carlson’s theorem

My question is on whether or not there exists some monotone strictly decreasing sequence of positive numbers $c_1>c_2>\ldots$ such that given any $f$ which is a uniformly bounded holomorphic ...
Ali's user avatar
  • 4,143
10 votes
0 answers
653 views

Eigenfunctions of the integral kernel $1/(x^2 + x'^2)$

My question seems elementary, yet I could not find the solution after working on and searching for several days... I'd like to find the eigenfunctions of a simple integral kernel: \begin{equation} \...
Yuli Nazarov's user avatar
10 votes
0 answers
657 views

“Taylor series” is to “Volterra series” as “Laurent series” is to _________?

Preamble My question is similar to an earlier MathOverflow question: “Taylor series” is to “Volterra series” as “Padé approximant” is to _________? which I just answered (hopefully my first ever ...
Nike Dattani's user avatar
10 votes
0 answers
845 views

Witt's proof of Gelfand-Mazur / Ostrowski's Theorem

Previously asked on Math Stackexchange without answers. Background: As sort of a hobby, Ernst Witt gave extremely short proofs for famous theorems. This question is about his six-line proof of the ...
Torsten Schoeneberg's user avatar
10 votes
0 answers
207 views

Projective tensor squares of uniform algebras

In discussion with a colleague recently (Jan 2017), $\newcommand{\AD}{A({\bf D})}\newcommand{\CT}{C({\bf T})}$ I was reminded that if $A(D)$ denotes the disc algebra and $\iota: \AD\to \CT$ is the ...
Yemon Choi's user avatar
  • 25.8k
9 votes
1 answer
608 views

Interpolation theory and $C^k$-spaces

Consider the Banach spaces $C^k(M)$ ($k=0,1,2,\dots$), consisting of $k$times continuously differentiable functions $f:M\rightarrow \mathbb{C}$ on a closed manifold $M$ (or just the torus if that ...
Jan Bohr's user avatar
  • 779
8 votes
3 answers
1k views

Can the topological algebra of analytic functions be endowed with a norm that defines the natural topology?

Right, so in my research in complex analysis I was puzzled by this question which may have a simple approachable answer that eludes me, but I am truly itching to find out and in need of it so I am ...
Don John Prep's user avatar
8 votes
3 answers
429 views

A density claim

Suppose that $g_k\in C([1,2])$, $k\in \mathbb N$ are continuous functions such that $\|g_k\|_{C([1,2])} \leq \epsilon^k$ for some sufficiently small $\epsilon>0$. Is the following claim true: If $f\...
Ali's user avatar
  • 4,143
8 votes
2 answers
887 views

Sum of the norm of polynomials

Let $\bar D$ denote the closed unit disc in the complex plane. Consider the function $f:\bar D\longrightarrow \mathbb{C}$, defined as $f(z)=z$ for all $z\in \bar D$. Let $n\in \mathbb{N}$. For $1\leq ...
User93709's user avatar
  • 355
8 votes
0 answers
251 views

Struggling with a proof in the preprint 'Hermitian geometry on resolvent set'

I have been struggling for awhile with a particular argument in the paper below. I posted the question first on MathSE, but I got no answers. I understand however that MO might be an overreach for ...
WeakMath's user avatar
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
7 votes
1 answer
370 views

Duality of $H^1$ and BMO

While proving that the dual of $H^1$ is $BMO$ in Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, page 143, Stein says that we have $\left\Vert g \right\Vert_{H^1} \...
abbyJeffers's user avatar
7 votes
3 answers
385 views

On what kind of condition of a compact set $K$ in the plane, $C(K)$ has a generator?

Let $K\subset \Bbb{C}$ be a compact subset of the complex plane, and let $C(K)$ be the space of all complex continuous functions on $K$. We say that $f\in C(K)$ is a generator of $C(K)$ when the set $...
Li Jingyang's user avatar
7 votes
1 answer
207 views

Convex Hull of univalent functions and Bieberbach Conjecture

Consider the class $S$ of univalent functions on the unit disk $D$ normalized so that $f(0)=0$ and $f'(0)=1$. Each function in $S$ satisfy the Bieberbach conjecture, that is the $n$-th coefficient in ...
user44316's user avatar
  • 185
7 votes
0 answers
745 views

What function space does holomorphic functional calculus give us?

Let $A$ be a unital Banach algebra, $U$ be an open subset of $\mathbb{C}$, and $A_U:=\{x\in A:\sigma(x)\subset U\}$. Holomorphic functional calculus says that any holomorphic function $f:U\rightarrow\...
Rand al'Thor's user avatar
6 votes
2 answers
347 views

Does there exist a framework for determining if a power series is "differentially algebraic"

It is a well studied problem to take a function $f$ expressed (usually expressed as a solution to a differential equation w/ some initial conditions) and ask if it has an "elementary closed form&...
Sidharth Ghoshal's user avatar
6 votes
1 answer
299 views

Continuity of eigenvectors

Let $\mathbb{C} \ni z \mapsto M(z)$ be a square matrix depending holomorphically on a parameter $z$ with the property that $\operatorname{dim}\ker(M(z)))=1$ for $z $ away from a discrete set $D \...
Sascha's user avatar
  • 536
6 votes
2 answers
240 views

Continuity of a differential of a Banach-valued holomorphic map

Originally posted on MSE. Let $U$ be an open set in $\mathbb{C}^{n}$ let $F$ be a Banach space (in my case even a dual Banach space), and let $\varphi:U\to F$ be a holomorphic map. I seem to be able ...
erz's user avatar
  • 5,529
6 votes
2 answers
672 views

Holomorphy of a function with values in a Hilbert space

Denote by $\mathbb C^\infty $ the Hilbert space $\ell^2 (\mathbb C)$. Fix $1\leq N,M \leq \infty$, and let $U$ be an open subset of $\mathbb C^N $. Following Mujica's book "complex analysis in Banach ...
Ervin's user avatar
  • 395
6 votes
2 answers
201 views

holomorphy in infinite dimensions (holomorphic families of operators)

Let $X$ be a Banach space (over $\mathbb C$), and let $\mathcal L(X)$ be its algebra of bounded linear operators. Let $U\subset \mathbb C^N$ be an open subset, and $f:U\to \mathcal L(X)$ a function ...
André Henriques's user avatar
6 votes
1 answer
182 views

Mittag-Leffler function

Let the Mittaq-Leffler function be defined by the expression $$ E_{\mu,\nu}(z) = \sum_{k=0}^{\infty} \frac{z^k}{\Gamma(k\mu+\nu)}\quad \text{$\mu>0$ and $\nu\in \mathbb R$}$$ Now let $n\in \mathbb ...
Ali's user avatar
  • 4,143
6 votes
1 answer
321 views

Derivatives of norm of vector-valued holomorphic functions

Let $G$ be a connected domain in $\mathbb{C}^{n}$, let $H$ be a Hilbert space and let $f,g:G\to H\backslash \{0\}$ be holomorphic (in my particular situation they are also injective, but I don't think ...
erz's user avatar
  • 5,529
6 votes
1 answer
291 views

Analytic maps on Banach spaces: analyticity upgrade

Consider the following problem. Let $E,F,G$ be real or complex Banach spaces, such that $F\subset G$ with continuous embedding. Let $U\subset E$ an open set and $$ f:U\to G $$ an analytic map, such ...
Lorenzo Pompili's user avatar
6 votes
1 answer
260 views

Do analytic functionals form a cosheaf?

Let $X$ be a complex-analytic manifold. Consider the sheaf of holomorphic functions $\mathcal{O}_X$ as a sheaf with values in the category of locally convex vector spaces. For $U\subseteq X$ open, we ...
Daniel Bruegmann's user avatar
6 votes
0 answers
220 views

Is the Taylor map continuous?

(Skip to the bolded theorem below for my question, if you'd like) Some context on asymptotic expansions and the Taylor map In the setting of irregular singularities of meromorphic connections on the ...
Brian Hepler's user avatar
5 votes
2 answers
330 views

Functional equation of bounded analytic functions

Let $\mathbb{D}:=\{z \in \mathbb{C}|~|z|<1\}$. Consider $f,~g \in H^{\infty}(\mathbb{D}):=\{f\in\text{ Hol}(\mathbb{D}): \|f\|_{\infty}:= \sup_{z \in \mathbb{D}} |f(z)| < \infty\}$ such that $f^...
Sherlok's user avatar
  • 149
5 votes
1 answer
353 views

Family of functions with prescribed derivatives

Suppose $f: \mathbb C \times (-1,1) \to \mathbb C$ is a smooth function that satisfies $f(0,t)=1$ for all $t\in (-1,1)$. Assume that for any $k\in \mathbb N$, any $z \in \mathbb C$ and any $t \in (-1,...
Ali's user avatar
  • 4,143

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