All Questions
Tagged with cv.complex-variables fa.functional-analysis
298 questions
1
vote
0
answers
86
views
Does there exist a bounded analytic function majorated by a given one?
Let $f$ be an element of the Hardy space $H^2$, i.e. $f$ is an analytic function on the unit disk such that $\sum|a_n|^2<\infty$, where $f(z)=\sum a_n z^n$. Assume also that $f\not\equiv 0$.
Is ...
- 5,529
4
votes
0
answers
117
views
Korovkin subset of $C(\mathbb{T})$
Let $K$ be a compact Hausdorff space and $A$ be a subset of $C(K)$. $A$ is said to be a Korovkin set if for every sequence $(T_n)$ of positive linear operators on $C(K)$, the condition $\|T_n(f)- f\|_\...
- 521
2
votes
0
answers
125
views
Sub Banach spaces (Banach algebras) of the disc algebra which are invariant under the differentiation operator
In this question the disc algebra $\mathcal{A}(\mathbb{D})$ is the Banach algebra of all holomorphic functions on the unit open disc $\mathbb{D} \subset \mathbb{C}$ which have a ...
- 356
1
vote
0
answers
87
views
An oscillatory integral estimate
Let $n \geq 3$ and consider two sequences of strictly monotone functions $\{\mu_l(t)\}_{l=1}^{n}$ and $\{\lambda_l(t)\}_{l=1}^n$ on the interval $[-1,1]$ with $\mu_l(0)=0$ and $\lambda_l(0)=1$ for all ...
- 4,115
2
votes
0
answers
119
views
Covariant derivative of the Monge-Ampere equation on Kähler manifolds
I am reading D. Joyce book "Compact manifolds with special holonomy" and I have some problems of understanding some computation on page 111, the first line in the proof of Proposition 5.4.6. More ...
- 21
4
votes
1
answer
212
views
Is the disk algebra a complemented subspace of the algebra of bounded analytic functions?
It is well known that the disk algebra (viewed as an algebra on the circle) is uncomplemented in $C(\mathbb T)$. What can be said about the pair
$(A(\mathbb D), H^\infty(\mathbb D))$?
- 687
1
vote
1
answer
596
views
A question about the proof of Riesz-Thorin interpolation theorem
I was reading the proof of Riesz-Thorin interpolation theorem in http://www.math.kit.edu/iana3/lehre/fourierana2014w/media/rieszthorinproof.pdf
and get stuck at the last step. We construct the complex ...
- 323
1
vote
0
answers
142
views
Affine algebraic variety as a set of common zeroes of holomorphic functions on ${\mathbb C}^n$
Let $V$ be an affine algebraic variety in ${\mathbb C}^n$, i.e. a set of common zeroes of some set $S$ of polynomials on ${\mathbb C}^n$:
$$
V=\{z\in {\mathbb C}^n:\ \forall p\in S\quad p(z)=0\}.
$$
...
- 7,426
1
vote
3
answers
207
views
Existence of solution to linear fractional equation
We consider the equation
$$ \sum_{j=1}^n \frac{\lambda_j}{x-x_j} =i$$
where $\lambda_j>0$ and $x_j$ are real distinct numbers.
I want to show that if $\lambda_k$ is small compared to the ...
user121558
6
votes
1
answer
259
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 ...
- 890
3
votes
0
answers
235
views
Chern number of projection-Topological magic in physics
I enclosed a computation from a well-known paper in the field of mathematical physics where the Chern number of the first Landau level is computed (the result claimed is $-1$) and the full paper can ...
- 31
4
votes
2
answers
186
views
Density of Lacunary Functions
I'm curious whether lacunary functions (functions in the complex plane that are holomorphic in some open ball about the origin, but cannot be analytically extended past that ball) are typical or the ...
- 435
5
votes
1
answer
850
views
$L\log L$ and Hardy space on the upper half plane
Set $\mathbb{T}$ the unit circle, $dm$ the Lebesgue measure on $\mathbb{T}$ and $\mathbb{C}^+=\left\{z\in \mathbb{C},s.t.\,\Im(z)>0 \right\}$ the upper half plane.
It is well-known that the Cauchy ...
- 51
10
votes
2
answers
737
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
...
- 3,738
4
votes
0
answers
275
views
Computing Bohr Radii
The Bohr radius $R$ for $\mathcal{H}(\mathbb{D})$ is defined as the radius $$R = \sup\limits_{0<r<1} \Bigl\{ r\ \Big|\sum\limits_{k=0}^{\infty}|a_k|r^k \leq |f|_\mathbb{D}, \text{ for all }f(z)=\...
- 3,209
2
votes
0
answers
219
views
Integral with product of two infinite sums
I am looking for references and results on integrals with product of two infinite sums:
$$S_1=\int_0^{\infty} \sum_{n=0}^{\infty} f(nx) \sum_{k=0}^{\infty} \overline{ g(kx)} dx$$
Above integral is ...
- 1,199
15
votes
1
answer
1k
views
Borel-Écalle re-summation and resurgence: criteria and results
This is about the theory of Borel-Écalle re-summation and resurgence, see Refs below.
This states that the perturbative series (say of the vacuum expectation value of an operator $\mathcal{O}$ in ...
- 10.5k
3
votes
0
answers
223
views
Sobolev space under Mellin transform
The Mellin transform is known to be an isomorphism see wikipedia
between $M:L^2(0, \infty) \rightarrow L^2(-\infty, \infty)$
where $$M(f):= \frac{1}{\sqrt{2\pi}}\int_0^{\infty} x^{-\frac{1}{2} + is} ...
user127612
5
votes
2
answers
279
views
Vector valued disc "algebra"
I am interested in a vector-valued form of the disc "algebra" (which in this setting is not in general an algebra, hence the scare quotes). Let $E$ be a Banach space, and let $A(\mathbb D,E)$ be the ...
- 18.7k
1
vote
1
answer
136
views
Conditions to obtain a real logarithm of a unitary unimodular complex matrix?
The problem statement is the following:
$$U=\exp\{iV\}$$
where $U$ is a unitary unimodular matrix of the following form:
$$U=\begin{bmatrix}u_1+iu_2&u_3+iu_4\\-u_3+iu_4&u_1-iu_2\end{bmatrix}...
- 113
10
votes
0
answers
844
views
Witt's proof of Gelfand-Mazur / Ostrowski's Theorem
Previously asked on Math Stackexchange without answers.
Background: As sort of a hobby, Ernst Witt gave extremely short proofs for famous theorems. This question is about his six-line proof of the ...
- 2,444
2
votes
2
answers
270
views
An inner product on the vector space $\mathbb{R}[x_1,\cdots,x_n]_m$
For any given integers $m,n\geq1$, let $\mathbb{R}[x_1,\cdots,x_n]_m$ be the vector space of homogeneous polynomials of degree $m$ in $x_1,\cdots,x_n$ over the field of real numbers $\mathbb{R}$. ...
- 1,997
1
vote
0
answers
86
views
An equality of inner products of holomorphic curves
The following is the main result in the paper by Vinnikov, Putinar, Alpay: A Hilbert space approach to bounded analytic extension in the ball, 2003, Communications on Pure and Applied Analysis. The ...
- 91
4
votes
0
answers
210
views
Inclusion of Hardy spaces
It is well-known that any convergence in $L^p$ for $p \in [1,\infty]$ implies convergence in $L^1_{\text{loc}}$ by Hölder's inequality.
It is also known that for $p>1$ it holds that $L^p(\mathbb R)...
1
vote
1
answer
220
views
How to prove that weighted Bergman space is separable.
Let $D$ be a bounded domain in $\mathbb{C}^n$ and $\varphi$ be a non-positive plurisubharmonic function on $D$. The weighted Bergman space $A^2(D,e^{-\varphi})$ is the space of holomorphic functions ...
- 11
4
votes
1
answer
233
views
Neumann DBAR problem with tempered distributions
It is well-known that the operator $$\frac{\partial}{\partial \overline{z}} : C^{\infty}(\mathbb{C}) \to C^{\infty}(\mathbb{C})$$ is surjective. (And it also works if we replace functions by Schwartz ...
- 1,017
1
vote
1
answer
167
views
Sampling set: relatively dense and uniformly discrete
The Paley-Wiener space of a domain $\Omega\subset\mathbb{R}^d$ is the set $$PW_\Omega:=\{f\in L^2(\mathbb{R}^d):\text{supp}\widehat{f}\subset\Omega\}.$$
We say that a discrete set $\Lambda\subset\...
- 319
3
votes
0
answers
182
views
Prove a certain function maps to upper half plane
Suppose $M$ is a bounded self-adjoint operator on space of complex valued functions on the real line $S_1=L^2(\mathbb{R},a(x)dx)$, where $a(x)$ is a nice real positive analytical function ( I have in ...
- 31
11
votes
1
answer
602
views
How do analysts think about functions with poles at all roots of unity?
In branches of algebra impinging on the enumeration of partitions, one often encounters formulas like
$$\prod_i \left( \frac{1}{1-q^i} \right)^{n_i}$$
for some integers $n_i$. E.g., with $n_i = 1$, ...
- 8,723
2
votes
1
answer
298
views
Regarding outer functions again
Consider the Hardy space $H^p, 0<p\leq\infty$ (defined here).
It is said that given any two outer functions $x_1$ and $x_2$ in $H^p$, there exists $a_1$ and $a_2$ in $H^\infty$ such that $a_1x_1=...
- 111
2
votes
1
answer
168
views
Regarding representation of an outer function
Theorem 2.1 in the book ‘Theory of Hp spaces by Peter. L Duren states that : Any function $f$ analytic on the unit disc belongs to the Nevanlinna class iff it is of the form $\frac{g}{h}$ where $g$ ...
- 111
4
votes
1
answer
253
views
Regarding outer functions
Please see the definition of Hardy spaces on the unit disc here. Let $0<p\leq\infty$. Let $f\in H^p$ with $\|f-1_e\|_p<1$ (Where $1_e$ Is the constant function one). Then is $f$ an outer ...
- 111
2
votes
0
answers
245
views
Dual space of functions of exponential type
The dual space of entire functions is known. It is the space of functions analytic around infinity with non-constant term, $\mathcal{O}^\infty_0$. The action of $F\in \mathcal{O}^\infty_0$ on an ...
- 503
3
votes
1
answer
125
views
Does uniform boundedness carry over from the non-negative real axis to closed sectors of $\mathbb{C}$ for analytic semigroups?
I'm trying to understand the proof of Theorem 2.5.6 (chapter 2.5) in Amnon Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer 1983.
For the direction (a)...
- 175
1
vote
1
answer
535
views
Incoherence of Fubini therorem with integral on Fourier series
I ask this question because of the apparent incoherence of the value of following integral:
$$I=\int_{0}^{1} \int_{0}^{\infty} \left|\sum_{n=1}^{\infty} f(nx) e^{2 i \pi n y} \right|^2 dx dy$$
...
- 1,199
1
vote
0
answers
82
views
Hankel operator with symbol a Blaschke product
If $B={\prod}_j \varphi_j$ is a Blaschke product (finite or infinite) of Blaschke factors $\varphi_j(w)=\frac{w-\alpha_j}{1-\overline{\alpha_j}w}$ with $|\alpha_j|>1$, is it true that the norm of ...
- 11
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 ...
- 149
0
votes
1
answer
185
views
Meromorphic solutions to Legendre's equation
I just saw the following question that was asked yesterday on math overflow on meromorphic solutions to ODEs
Although, I understand the answers and comments to the questions, I did not understand how ...
- 501
2
votes
2
answers
258
views
Meromorphic extension of solutions to ODEs
I encountered the following question in my studies:
Let us assume we have a real anlaytic solution to an ODE on $\mathbb{R}$ of Schr\"odinger type
$-\psi''(x)+V(x)\psi(x)=\lambda \psi(x)$
but we ...
- 167
2
votes
0
answers
379
views
Is this double integral of Fourier series always real?
Consider $f(x)$ a function from $\mathbb{R^+}$ to $\mathbb{C}$ such that $f(x) \sim_0 x$ and $\int_{0}^{\infty} f(x) dx=\int_{0}^{\infty} x^2 f(x) dx=0$
Can we demonstrate that following integral is ...
- 1,199
5
votes
1
answer
243
views
Deformation of the Plücker coordinates
Let $M_{2,4}(\mathbb{R})$ be the set of real $2\times4$-matrices of rank $2$. For any $A\in M_{2,4}(\mathbb{R})$ and $1\leq i<j\leq 4$, let $p_{ij}$ be the corresponding $2\times 2$-minors of $A$. ...
- 93
2
votes
0
answers
88
views
Link between subharmonic and subanalytic functions
Consider $\Omega$ an open set of $\mathbb{C}$ and $f : \Omega \to \mathbb{R}$ a $C_{\infty}$-function. The following two definitions are well-known (at least the first one) but I prefer to recall them ...
- 1,017
5
votes
1
answer
390
views
Is this closed subspace of Fréchet space complemented
In the hope of completing the rich tapestry of complemented (or not) topological vector subspaces, I would like to know (maybe it is immediate for specialists)
whether the space of analytic functions ...
- 3,738
3
votes
0
answers
187
views
Families of unbounded operators
Let $H$ be a Hilbert space, $X$ a topological space, and $\{A_t\}_{t\in X}$ a continuous family of bounded, invertible operators on $H$. Continuous here in the sense that the corresponding map $X\...
- 331
3
votes
2
answers
470
views
inner product on matrix spaces of multivariate polynomials?
Let $H_{n,d}=\mathbb{R}_d[x_1,..,x_n]$ be the space of $n$-variate homogeneous degree $d$ polynomials, $D=D^\top\in \mathbb{N}^{m\times m}$ a symmetric $m\times m$ matrix. Consider the space $P_D$ of ...
- 14k
3
votes
1
answer
195
views
density of holomorphic functions in vertical strips
Define H(a) to be the space of holomorphic functions $f(z)$ on $S_a:=\{z:|\Re z| < a\}$ with
$$ ||f||^2_a:=\int_{S_a} |f(x+iy)|^2 (1+|x+iy|)^{100} dx \, dy < \infty. $$
Two questions:
Is H(2) ...
- 151
5
votes
1
answer
594
views
Explicit Paley-Wiener function
By a Paley-Wiener function I mean a function $f(z)$ that is the Fourier image of a test function. Equivalently, by Paley-Wiener theorem, $f(z)$ is an entire function that is of rapid decay on the real ...
- 1,959
2
votes
1
answer
113
views
Norm of vector-valued holomorphic functions
Let $G$ be a connected simply connected domain in $\mathbb{C}^{n}$, let $H$ be a Hilbert space.
Q1. Which functions $F:G\to(0,+\infty)$ are such that there is a holomorphic $f:G\to H\backslash ...
- 5,529
2
votes
0
answers
136
views
To find a positive function with compact spectrum
Let
$e_1=(0,1)^T$,
$$
S=\left\{x\in \mathbb{R}^2\Big| \frac{|\langle x, e_1\rangle|}{|x|}>\delta>0\right\},
$$
is a cone in $\mathbb{R}^2$.
I want to find a non-trivial smooth function ...
- 29
6
votes
1
answer
319
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 ...
- 5,529