0
votes
1answer
16 views

an example of a function that is integrable m times and has first deriv cont and bdd

could someone provide some example of a function that can be integrated m times (for m between 1 and inf) AND has first deriv cont and bounded.
1
vote
2answers
33 views

Geometrical Meaning of derivative of complex function

What's the geometrical meaning of f'(z) in complex analysis, as we know in real analysis f'(x) has meaning ie. Slope of curve or gives max/ min. But what does derivative f'(z) has geometrical meaning ...
0
votes
1answer
19 views

Linear indepedent holomorphic functions

Suppose you have a given set of holomorphic functions $e_\alpha(z)=\exp(\alpha_1z_1+\dots+\alpha_nz_n)$ for different $\alpha=(\alpha_1,\dots,\alpha_n)$ in an open set of $\mathbb{C}^n$. How can I ...
2
votes
1answer
54 views

Application of Riesz representation theorem

Suppose the following situation. We have linear functional $l$ on the space $H(\mathbb{C}^n)$ of entire function and wish to find a representation for $l$ with integration against a complex Borel ...
0
votes
1answer
35 views

One-sided total derivative

Given a function from half space into euclidean space: $f:\mathbb{H}^m\to\mathbb{R}^n$ Suppose its one-sided limit exists at a specific point: $\lim_{\mathbb{H}^m\owns v\to 0}\frac{1}{\lVert ...
1
vote
0answers
33 views

Differentiabilty of certain convolution

Let $\Gamma$ be a smooth closed curve. Suppose $f\in L^2(\Gamma,ds)$ and $g$ is defined everywhere on $\mathbb{C}$ with compact support. Moreover $\frac{dg}{dz}$ exists everywhere but point $z=0$. ...
5
votes
1answer
37 views

Norm of a functional on square integrable harmonic functions

Let H be the Hilbert space of square integrable (real) harmonic functions on the unit disk of the complex plane. I want to find the norm of the linear functional $$h\mapsto h_x(0)$$ Here is my proof ...
3
votes
1answer
99 views

The derivative of harmonic function at origin is a bounded linear functional

The following problem is the 5th problem in the qualifying exam of UCLA (spring 2013). Let $\mathbb{D}=\{(x,y):x^2+y^2<1\}$ and let us define a Hilbert space $$H:=\{u:\mathbb{D} \rightarrow ...
1
vote
0answers
17 views

Analytic family of operators?

If $E_{\lambda}, F_{\lambda}$ are two families of complex Hilbert spaces and $L_{\lambda} : E_{\lambda} \rightarrow F_{\lambda}$ is a family of bounded linear operators, where $\lambda$ is a complex ...
2
votes
1answer
30 views

Why is $F_\phi$ defined on the whole disk

This is a question about a proof on page 97 in these lecture notes. In exercise 13, I don't understand On the hand, $F_\phi$ is defined on the whole open disk $D$ Why is $F_\phi$ defined on ...
0
votes
1answer
49 views

compactness in the space of analytic functions

I am always getting confused by the idea of compactness so I would like some help to see whether a set is compact. (I need this to prove existence of a solution of a map) So let $D\in\mathbb{C}$ be ...
1
vote
0answers
44 views

What is a good book that focuses on the applications of complex analysis and spectral theory?

My research involves a great deal of complex analysis and spectral theory, and I always feel a bit flustered when non mathematicians ask me what I study. It's hard to explain the math in layman's ...
1
vote
0answers
26 views

Real-valued Irreducible Representations of Lie Groups

I'm interested in the real-valued irreducible representations of a number of Lie groups. For concreteness I'll use the group $M(2)$ of Euclidean motions, which can be parameterized as follows: $$ g(t, ...
0
votes
1answer
44 views

Show that $\ c_X(p,q) \le d_X(p,q)$, for $ p, q \in X$

Update I'm trying to show the Corollary, but I have stuck...That is: For any complex space $X$, we have: $$\begin{align} (1).\ c_X(p,q) &\le d_X(p,q),\ \text{for}\ p, q \in X \\ (2).\ ...
0
votes
0answers
32 views

The Kobayashi pseudo - distance $d_X$ and the Carathéodory pseudo - distance $c_X$

I'm studying the Kobayashi pseudo - distance $d_X$ and the Carathéodory pseudo - distance $c_X$. And I have trouble when I try to show $4$ properties of $c_X$ in my textbook. {It doesn't have ...
2
votes
1answer
74 views

Extremum of functional of a complex function

consider functional $E$ defined by $$E[z]=\int F(x,z(x))dx$$ where $F$ is a complex-valued nonlinear function. How can we find the function $z(x)$ so that $$G=|E|^2=EE^*=\iint ...
2
votes
1answer
83 views

Behavior of the resolvent near the boundary of the spectrum

My question is, in some sense, a continuation of the question below. Isolated singularities of the resolvent Suppose $T\in B(H)$ has no eigenvalues, pick $x\in H$, $x\neq 0$, and consider the ...
3
votes
2answers
75 views

Isolated singularities of the resolvent

Let $T$ be a bounded operator on $l_2$ such that there exists $\mu$ in the spectrum of $T$ which is an isolated point of the spectrum. We know that for any $x\in l_2$ the resolvent function ...
5
votes
2answers
56 views

Trying to show that $z \mapsto f_z : \mathbb{C} \to L^1(\mathbb{R})$ is complex differentiable where $f_z(x) = e^{-(x+z)^2}$

Let $g$ be the entire function $g(z) = e^{-z^2}$. Note $g$ is integrable along every horizontal line. For each complex number $z \in \mathbb{C}$, define $f_z : \mathbb{R} \to \mathbb{C}$ by $f_z(x) = ...
1
vote
1answer
51 views

Fourier transform of $\frac{d}{dt}\ln\frac{1}{it}$

I'd like to proove the identity $$\mathcal{F}\left(\frac{d}{dt}\ln\frac{1}{it}\right)=2i\pi H$$ with $H=\mathbb{I}_{\mathbb{R}^+}$ ie the Heaviside step function, $\mathcal{F}$ denote the Fourier ...
2
votes
1answer
49 views

Compact Function Set

If a uniformly equicontinuous family of functions is analytic on an open disk in the complex plane, it has compact closure by Montel's theorem (and Arzela-Ascoli). Is it possible that this set is ...
1
vote
1answer
26 views

Summation of the Bergman kernel at two distinct points is constant?

Let $\Omega$ be a bounded simply connected domain in $\mathbb{C}.$ Let $K(z,w)$ denotes the Bergman kernel of $\Omega.$ Let $w_1,\,w_2$ be two distinct points in $\Omega.$ I'm looking for a domain ...
1
vote
1answer
65 views

In a Banach space X, its two Schauder bases have the same cardinal number?

The definition of Schauder basis is, there exist a set family F(whose cardinal number can be finite countable or uncountable), s.t. any x in X could be uniquely expressed countalbe linear combinations ...
2
votes
1answer
56 views

Holomorphic functions on the product of open sets.

Is it true that $$ \mathcal H(\mathrm U \times \mathrm V) \simeq \mathcal H(\mathrm U) \widehat{\otimes} \mathcal H(\mathrm V) $$ for open two open affine sets $\mathrm U$ and $\mathrm V$? Edit: I ...
3
votes
1answer
35 views

A question on a sequence in a Banach algebra [duplicate]

If $\{u_{k}\}_{k=1}^{\infty}$ is a sequence in an Banach algebra (and more specifically, in the set of all the bounded linear operators of a Banach space $X$). If ...
0
votes
1answer
26 views

determine the maximal open subset $G\subset\mathbb{C}^2$ such that the series $f$ converges in $\mathcal{O}(G)$ in the topology of Frechet space.

Consider $f(z_1,z_2)=\sum\limits_{j=0}^\infty(z_1+z_2)^j$,determine the maximal open subset $G\subset\mathbb{C}^2$ such that the series $f$ converges in $\mathcal{O}(G)$ in the topology of Frechet ...
2
votes
1answer
68 views

Additional hypotheses needed for the converse of $\mathscr F \subset H(G)$ normal $\implies$ $\mathscr F ' = \{f' : f \in \mathscr F\}$ normal?

Problem Show that if $\mathscr F \subset H(G)$ is normal then $\mathscr F' = \{f' : f \in \mathscr F\}$ is also normal. Is the converse true? Can you add something to the hypothesis that $\mathscr ...
0
votes
0answers
35 views

Problem with the Cauchy-Green transform

Let $u \in C_c(\mathbb C)$. Then it's Cauchy-Green transform $$ \tilde u(z) = -\frac{1}{\pi} \int\limits_{\mathbb C} \frac{u(\xi)}{\xi-z} d\xi_R d\xi_I, \quad \xi = \xi_R + i \xi_I, $$ is ...
2
votes
1answer
51 views

how to show f is bijective?

Suppose that the set-mapping $f:X\rightarrow Y$ of one-dimensional domains of $\mathbb{C}$ induces an isomorphism $f^0:\mathcal{O}(Y)\rightarrow \mathcal{O}(X)$ defined by $g\mapsto g\circ f$ of ...
1
vote
0answers
26 views

Find Limit of Complex integration

How should we prove that $\int_{\mathbb{C}}\frac{|G(w)|}{|z-w|}dA(w)\longrightarrow0$ as $|z|\longrightarrow\infty$where $G(z)\in L^{1}(\mathbb{C})$ and $|G(z)|\leq\frac{C_{0}}{|z|^{1+\varepsilon}}$ ...
1
vote
1answer
37 views

Is holomorphic functions on (0, 1) (vanishing at endpoint) dense in $C_0((0, 1))$?

Here is my argument, please let me know if it works or not. By Stone-Weierstrass Theorem (Complex Version), functions in $C_0((0, 1))$ can be uniformly approximated by polynomials in z and $\bar{z}$ ...
0
votes
1answer
54 views

An application of Runge's Theorem to approximate analytic functions by polynomials

Apply the following form of Runge's Theorem: if $X\subset \mathbb{C}$ is an open subset,and if $\mathbb{C}\setminus X$ is connected,then $\mathbb{C}[z]$ is dense in $\mathcal{O}(X)$ in the topology of ...
1
vote
0answers
42 views

Constructing an L2 function from an entire function bounded on R

I have an entire function $f(z)$ of exponential type $\tau\geq0$ that is bounded on $\mathbb{R}$ and zero at every member of the complex sequence $\{\lambda_n\}$. What I want is an entire function of ...
1
vote
1answer
30 views

Need an example on the topology of compact convergence

Find polynomials $f,g\in \mathbb{C}[z]$ and a $K\subset\subset \mathbb{C}$ such that \begin{equation} \|fg\|_K<\|f\|_K\|g\|_K \end{equation} where $\|f\|_K:=\sup_{t\in K}|f(t)|$. I have tried ...
2
votes
2answers
80 views

Show that the operator is bounded in $L_p$

Consider the operator $C$, acting on functions $f$ on the unit circle $S^1 = \left\{ z \in \mathbb C \mid |z| = 1 \right\}$ by the rule $$ (Cf)(z) = \frac{1}{2\pi i} ...
1
vote
0answers
32 views

Composition of analytic functions is analytic in a general setting, and are they continuous?

Regarding the notion of analyticity discussed in this setting: A possible equivalence for holomorphicity I wonder if this is truly the correct definition (even though it is from Dunford-Schwarz) An ...
0
votes
0answers
23 views

A possible equivalence for holomorphicity

Let $X$ and $Y$ be Complex Banach spaces with $U$ an open subset of $X$, and $f:U\rightarrow Y$. We say that f is analytic/holomorphic if for every $z_1, ... z_n \in X$ we have that the mapping $a_1, ...
2
votes
1answer
92 views

Question related to the spectrum of a bounded operator

If $A$ is a bounded linear operator on a Banach space $X$ and $\lambda\in \sigma(A)$, is it true that for all $\epsilon>0$, there is $ x\in X$ and $||x||=1$ such that $$ ||(A-\lambda I) x|| ...
2
votes
0answers
84 views

Explicit example of a function with Fourier transform in $C_0^\infty(\mathbb{R})$

Is anyone aware of an explicit example of a (Schwartz, real-analytic, extending to an entire function with suitable decay properties along imaginary directions as for the Paley–Wiener-Schwartz ...
0
votes
1answer
52 views

proof of lemma 10-8, In functional analysis book of Rudin page 232

In functional analysis book of Rudin page 232, proof of lemma 10-8 We have a function $ h_r(\lambda)= \frac{r^2 g(\lambda)}{z^2(2r-g(\lambda))} , \lambda \in \mathbb{C} $ and $ g(\lambda)$ is an ...
1
vote
2answers
106 views

Sequences and Contraction of a fixed point

Suppose that $g:\mathbb R \to $$\mathbb R$ is a contraction. Then $g$ has a unique fixed point $c$ and that for any number $x_0$, the sequence $x_0, x_1, x_2,\ldots$ given by $x_n = g(x_{n-1})$. ...
0
votes
1answer
29 views

Image of innerproduct unordered field?!

This might be totally stupid, but I have a problem regarding IPS and NLS. My problem is that as I understand, one does not assume that the image of the innerproduct is an ordered field..? In the ...
1
vote
2answers
68 views

A question about analytic function on the neighbourhood of spectrum sp(x).

I am reading a paper about operator theory. But i meet with a problem. Let x be a element of Banach alegebra A and f be a arbitrary function analytic on the neighbourhood of sp(x) (sp(x) is the ...
3
votes
1answer
63 views

Bounded Holomorphic Function - Banach Space?

Can someone, help me in this question, please? Let $U\subset\mathbb{C}$ be open set and $H_\infty(U)=\{f:U\to\mathbb{C}:f\text{ is bounded and homolorphic}\}$. Show that $H_\infty(U)$ is a closed ...
2
votes
2answers
149 views
0
votes
1answer
80 views

Bound of $\log \det$

I want to find a bound to the function $$R(d_i, ...
0
votes
1answer
62 views

Showing that a set is balanced but its interior is not

I'm having some difficulty visualizing a particular set for a functional analysis homework problem. That is, let $B= \{(z_1, z_2) \in \mathbb{C}^2: |z_1| \leq |z_2|\}$. I'm asked to show that $B$ is ...
1
vote
1answer
44 views

Dealing with partial derivatives in a function space

Please read the following details below. Question: I want to show now that if $r>s>0,f \in F_s (\Omega), $ and $u \in F_r (\Omega)$, then for any $i$, $$f \frac{\partial u}{\partial z_i} ...
1
vote
1answer
110 views

Cauchy Derivative Estimates for entire functions with a bound.

The problem statement: Assume $f$ is an entire function and that there is an $n \in \mathbb{N}$ and a $C < \infty$ such that for $z \in C$ $$|f(z)| \le C ( 1+|z|^n)$$ Also assume that $f$ is never ...
3
votes
1answer
37 views

Analytic continuation of one parameter subgroup: group property preserved?

Let $(\mathcal{A},\alpha)$ be a C* dynamical system, i.e. $\mathcal{A}$ is a unital C*-algebra and $\{\alpha_t\}_{t\in \mathbb{R}}$ a strongly continuous one-parameter group of *-automorphisms. For ...