All Questions
Tagged with cv.complex-variables fa.functional-analysis
298 questions
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 ...
- 25.8k
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 ...
- 91.8k
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 ...
- 18.7k
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 ...
- 91.8k
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 ...
- 91.8k
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 $A$. ...
- 2,784
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 ...
- 25.8k
2
votes
0
answers
89
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
2
votes
0
answers
70
views
The $n$-th derivative for $z$ at $z = 0$ of $F(a,b)(z)$ is no less than that of $F(\frac{a+b}{2},\frac{a+b}{2})(z)$ [closed]
The proposition the OP wants to prove is incorrect. --Aug 8, 2017
Can we find a elegant way to prove that the $n$-th derivative for $z$ at $z = 0$ of $F(a,b)(z)$ is no less than that of $F(\frac{a+...
- 1,551
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
471
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 ...
- 8,597
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
595
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 ...
- 91.8k
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
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 ...
- 91.8k
6
votes
1
answer
320
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 ...
- 91.8k
3
votes
0
answers
89
views
Trace of a weighted composition operator on Bergman space
I am reading a series of papers by Pollicott, Jenkinson and coauthors which make use of the following type of result:
Theorem: Let $\mathbb{D} \subset \mathbb{C}^d$ be a bounded, connected open set. ...
- 6,206
5
votes
0
answers
104
views
On the embedding of manifolds into infinite-dimensional spaces
Let $X$ be a (connected, finitely dimensional) topological/smooth/complex manifold and let $i$ be a weakly continuous/continuous/smooth/holomorphic map from $X$ into the dual $F^{*}$ of a real or ...
- 5,529
4
votes
1
answer
155
views
For Hilbert spaces, does weak analyticity with respect to a dense subspace of functionals imply analyticity?
Let $i : X \hookrightarrow Y$ be a dense embedding of complex Hilbert spaces.
Let $f : \mathbb{D} \to X$ be a function, such that $i \circ f$ is holomorphic ($\mathbb{D}$ is the open unit disk). Is ...
- 16.4k
1
vote
1
answer
130
views
A problem on completeness of a specific space of complex functions
Let $C(D) \cap H(\bar D)$ denote the inner product space of functions these are analytic in unit disk $D$ and continuous in $\bar D$, equipped with the inner product $(f,g)= \frac{1}{2 \pi} \int_{0}^{...
- 91.8k
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 ...
- 3,146
0
votes
1
answer
150
views
Solutions to Schrödinger equation parameter dependence
This is somewhat unrelated to what I normally do in mathematics, which is why it may be obvious to some of you, but I was puzzled by this:
If we look for classical solutions on $[0,1]$ to
$$-y''(x) =...
- 305
3
votes
1
answer
167
views
Recovering residue using local real information
Let $f(z)$ be defined by a Laurent series at z = 0 with real coefficients. In particular, $f(x) \in \mathbb{R}$ for $x \in \mathbb R$.
Compute the residue of $f(z)$ at z = 0 using just the ...
- 91.8k
1
vote
2
answers
274
views
$\mathcal P(K)=\mathcal R(K)$ iff $\Bbb C\backslash K$ is connected
Let $K$ be a compact set in $\Bbb C$. Let $\mathcal P(K)$ be the closed algebra generated by polynomials on $K$ and $\mathcal R(K)$ the closed algebra generated by rational functions without poles in $...
- 2,154
1
vote
0
answers
80
views
What is the character space of $\mathcal P(K)$?
Let $K$ be a compact subset of $\Bbb C$. Let $\mathcal P(K)$ be the closed algebra generated by the complex polynomials on $K$.
What is the character space $\Phi_{\mathcal P(K)}$ of $\mathcal P(K)$?...
CommunityBot
- 1
3
votes
2
answers
457
views
Integrality of complex infinite series
Let $(a_n)$ be a (double-sided) sequence of complex numbers satisfying
$$\sum_{\mathbb{Z}}\vert n\vert\,\vert a_n\vert^2<\infty, \tag1$$
$$\sum_{\mathbb{Z}}a_n\bar{a}_{n+k}=\delta_0(k), \qquad \...
- 43.2k
1
vote
0
answers
128
views
determine when $e^{ikx}$ can be boundary value of a holomorphic function
Assume that $\Gamma=\{x+if(x): x\in \mathbb{R}\}$ is a graph, separating $\mathbb{C}$ into two connected components. Let's denote the one below $\Gamma$ by $\Omega$.
My question is, for what curves $...
- 24.2k
6
votes
2
answers
671
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 ...
CommunityBot
- 1
3
votes
0
answers
246
views
Inverse problem for negative moments
Let $D$ be a bounded simply connected domain in the plane, bounded by a smooth closed curve $\partial D$. Moreover$D$ contains the origin. Assume that all negative complex moments w.r.t arc-length ...
CommunityBot
- 1
2
votes
0
answers
627
views
Is every bijective analytic map bi-analytic?
Suppose that
$E$ and $F$ are complex Banach spaces and $U\subset E$ and $V\subset F$ are open subses.
$f\colon U\to V$ is analytic
$f\colon U\to V$ is bijective
Is $f$ bi-analytic? (i.e. is its ...
- 131
1
vote
0
answers
84
views
extension for a complex operator
Let be $\lambda>0$. Put
$$ L_{\lambda}=\Big[-\frac{\partial^{2}}{\partial z \partial \overline{z}}+\lambda^{2}|z|^{2} +\lambda\Big(\overline{z}\frac{\partial}{ \partial \overline{z}}-z\frac{\...
- 25.8k
3
votes
1
answer
305
views
The spectrum and the tangent space of the algebra of holomorphic functions on a Stein manifold
Let $A$ be a Fréchet algebra over ${\mathbb C}$, and let us call the spectrum ${\tt Spec}[A]$ of $A$ the set of all characters, i.e. continuous multiplicative linear functionals $s:A\to{\mathbb C}$, ...
- 7,426
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 ...
- 16.4k
2
votes
0
answers
154
views
The boundedness of an entire function along the imaginary axis
I am looking for an answer in the affirmative or the negative concerning the asymptotics of an entire function. The question, relatively simple to state, has proven very taxing to solve and requires ...
- 24.2k
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 ...
- 9,486
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 $...
CommunityBot
- 1
3
votes
2
answers
205
views
Bergman norm on a bigger domain
Let $D$ be a unit disc (I am actually interested in a much more general setting, but let's start with explicit examples). Let $E$ be an open subset of $D$. Consider the functional on the space of all ...
- 27
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:...
- 24.2k
1
vote
1
answer
137
views
When is a homogeneous polynomial an inner function on the unit torus?
What is the necessary and sufficient condition(s) for a homogeneous polynomial on torus(bidisk) to be an inner function? More precisely I want to know when absolute value of a homogeneous polynomial ...
- 2,361
0
votes
1
answer
226
views
Subspaces of $H^{\infty}(\mathbb{D})$ which contains a nontrivial weak* closed subalgebra
Let $H^{\infty}(\mathbb{D})$ denotes the Banach space of bounded holomorphic functions in the unit disc. Consider the weak* topology on $L^{\infty}(\mathbb{T})$
that it inherits as the dual of $L^{1}(\...
CommunityBot
- 1
2
votes
1
answer
244
views
Is the set of entire functions Borel in the space of analytic functions?
$\def\bbR{\mathbb R}\def\bbC{\mathbb C}\def\scrT{\mathscr T}\def\ssp{\kern.4mm}
$More specifically, I ask whether $S$ be a Borel set in the topological space $(\Omega,\scrT)$ in the following ...
- 3,584
0
votes
1
answer
128
views
On a theorem by Mooney and Khavin on the weak sequential completeness of the predual of $H^\infty(\mathbb{D})$
There is a theorem by Mooney http://msp.org/pjm/1972/43-2/pjm-v43-n2-p.pdf#page=185 and independently proved by Havin which says that the predual of $H^{\infty}(\mathbb{D}),$
$L^{1}/H^{1}_{0}$ is ...
- 2,361
1
vote
1
answer
534
views
Stone-Weierstrass Theorem, polynomial interpolation, divided difference in complex plane
Setting:
Let $\Gamma$ be a simple smooth($C^\infty$) curve in $\mathbb{C}$ parametrized by the injective map $\gamma:[0,1] \to \mathbb{C}$.
Assume $f$ is a function defined on $\Gamma$ s.t. $f$ is ...
- 398
4
votes
0
answers
715
views
Can one integrate around a branch-cut?
How meaningful is it to try to integrate around the branch-cut of a function?
For example lets say I have the function $\log(z^2+a^2)$ for $a>0$ and I choose my branch-cuts to be starting at $\pm ...
- 1,893
3
votes
0
answers
175
views
polynomial relations between modular functions
$\newcommand{\Qbar}{\overline{\mathbb{Q}}}$
We define a modular function to be a meromorphic modular form of weight 0 for some subgroup (not necessarily congruence) $\Gamma\le\text{SL}_2(\mathbb{Z})$ ...
CommunityBot
- 1
1
vote
0
answers
109
views
Zeros of functions constituting a Riesz-basis for the Paley-Wiener space
I have a short question which first requires some slightly elaborate definitions:
Let $(e_n)$ be a Riesz-basis for a Hilbert space $\mathcal{H}$ with biorthogonal basis $(g_n)$, i.e. $\langle e_m, ...
- 11
0
votes
1
answer
730
views
Reproducing Kernel of a RKHS of continuous functions may not be continuous in two variables together
Let $\mathcal{K}$ be a Hilbert Space of continuous functions on some topological space, where point evaluations are continuous linear functional on $\mathcal{K}$.
That is $\mathcal{K}$ is RKHS, ...
- 3,085
0
votes
0
answers
109
views
solutions of elliptic linear pde depending analytically on a parameter
Fix $ \Omega$ a bounded smooth domain in $ R^N$ and suppose $0<w(x)$ is a smooth solution of $ -\Delta w(x)=w(x)^2$ in $ \Omega$ with $ w=0$ on $ \partial \Omega$ (were are assuming $2< \frac{N+...
- 1,385
2
votes
0
answers
156
views
When does analytic in the operator norm imply analytic in the trace class norm?
This is a crosspost from MSE. It's been up there for a few weeks now. A 200 rep bounty yielded no results (or even comments). I'm hoping someone here has some helpful ideas. See this post for the ...
CommunityBot
- 1
0
votes
1
answer
152
views
When can two Cauchy transforms intersect?
Given two polynomials $p$ and $q$ over reals and being guaranteed that both have all roots real I want to know if there is any characterization of the solutions of the equation $\frac{p'}{p} = \frac{q'...
- 54.2k