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Holomorphic functions of certain blow up at origin

Suppose that $D=\{z\in \mathbb C\,:\, |z|\leq 1\}$ and let $f$ be holomorphic on $D\setminus\{0\}$ such that $|f(z)|\leq e^{\frac{1}{|z|}}$ for all $0<|z|\leq 1$ and assume additionally that $\lim\...
Ali's user avatar
  • 4,143
1 vote
0 answers
147 views

integral over the unit sphere of $\Bbb C^n$

Please, is there a way to calculate this integral $$\int_{S_{2n-1}} \frac{e^{a \langle z, \zeta \rangle}}{|z - \zeta|^{\beta}} \, d\sigma(\zeta)$$ where $ z $ is a fixed point in the complex unit ball ...
zoran  Vicovic's user avatar
2 votes
0 answers
70 views

Is the hypothesis "uniformly equivalent" needed?

I am reading S. Shimorin's paper titled Complete Nevanlinna-Pick property of Dirichlet-type spaces. My question concerns Lemma 2.3. which is as follows: Assume $\mathscr{H}$ is a Hilbert space of ...
ash's user avatar
  • 151
0 votes
1 answer
119 views

Nonstationary phase method for oscillatory integral

I want to approximate an integral of the form $$\int_a^bf(t)e^{ig(t)}dt,$$where $f(t)$ is smooth, $g(t)$ is real-valued and smooth. The stationary phase method says that if $t_0\in [a,b]$ is such that ...
charlie_beck's user avatar
0 votes
0 answers
146 views

On the pointwise limit of a sequence of analytic functions

I have been confused with this problem for weeks now. Suppose I have Banach spaces $E$ and $F$ and a sequence of functions $f_{n}: U \subset E \to F$, where $U$ is open and nonempty. Let $x \in U$ be ...
InMathweTrust's user avatar
1 vote
0 answers
71 views

Integral formula of quantum dilogarithm

In the paper"Level N Teichmüller TQFT and Complex Chern-Simons Theory" arXiv:1612.06986, the authors study the quantum dilogarithm function: \begin{equation} \mathrm{D}_{\rm b}(x,n)=\prod_{...
color's user avatar
  • 109
0 votes
0 answers
57 views

Double-periodic functions with (possible) poles

Consider the set of double-periodic function $f:\mathbb C/(\mathbb Z+i \mathbb Z) \setminus \{z_0\} \to \mathbb C$, where $z_0$ is a fixed point inside $\mathbb C/(\mathbb Z+i \mathbb Z),$ that have a ...
António Borges Santos's user avatar
1 vote
0 answers
38 views

About Carleson measures on the Hardy space on the bidisc

I have been reading the paper "Carleson Measures in Hardy and Weighted Bergman Spaces of Polydiscs" by F. Jafari and there are a few things that going on that I am not entirely convinced of. ...
an_ordinary_mathematician's user avatar
2 votes
0 answers
179 views

Analytic continuation of $\int_V (f(x_1,\cdots,x_n))^s dx_i$

Let $V$ be an $n$-dimensional simplex, let $f(\boldsymbol{x}) = f(x_1,\cdots,x_n)\in \mathbb{C}[x_1,\cdots,x_n]$ be a product of linear polynomials that is non-zero in interior of $V$. Also let $E(\...
pisco's user avatar
  • 528
4 votes
0 answers
148 views

Some questions on Hardy's spaces

In the paper http://www.numdam.org/item/CM_1976__33_3_261_0.pdf, the authors have asked in Page 285 whether the Hardy space $H^p$ embeds isometrically into the Hardy space $H^q$ for $1\leq q<p<...
A beginner mathmatician's user avatar
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0 answers
78 views

What does analytic uniformly in $s$ mean?

Suppose I have a complex vector space $V$ with finite basis $\{e_{1},...,e_{s}\}$. Then, I can consider the algebra $\mathcal{U}$ of formal polynomials on the variables $e_{1},...,e_{s}$. Suppose ...
InMathweTrust's user avatar
1 vote
1 answer
65 views

Reference dual Dirichlet space $D^1$

Let $\mathbb{D} = \{ z \in \mathbb{C} : |z| < 1 \}$ be the unit disk. The Bergman space $A^1 = A^1(\mathbb{D})$ is the Banach space of holomorphic functions on $\mathbb{D}$ such that $$ \|f\|_{A^1} ...
Scottish Questions's user avatar
0 votes
0 answers
36 views

Derivate involving Bessel function of second type

Let. $$f := (x, y) \mapsto \text{BesselK}(1, c \cdot (a - b \cdot (x + y))) \cdot \exp(c \cdot b \cdot (y - x))$$ Is there a close formula for this $$\frac{\partial^{m+n}}{\partial y^m \partial x^n} f(...
Ryo Ken's user avatar
  • 109
3 votes
0 answers
116 views

On a functional equation of Mahler?

Recently, I was trying to introduce the concept of natural boundaries to a fellow math student, and what greater way to do this than using an example? In particular, I tried to use as an illustration, ...
Prelude's user avatar
  • 131
2 votes
0 answers
95 views

On analytic functions on the complement of a curve without jump across the curve almost everywhere

Question. Suppose $f$ is an analytic function on $\mathbb C\setminus\mathbb R$ and assume that the boundary values of $f$ from above and below the real axis (denoted $f_\pm$ respectively) exist almost ...
RandomWalk123's user avatar
1 vote
1 answer
152 views

SOT and WOT convergence of Toeplitz operators

For the Hardy space $H^2$, every $\phi \in L^\infty (\mathbb T)$ induces a bounded Toeplitz operator $T_\phi$ on the Hardy space and $\lVert T_\phi \rVert = \lVert \phi \rVert _{\infty}$. Consequently,...
ash's user avatar
  • 151
2 votes
1 answer
99 views

A question on Bloch functions

Let $\mathcal{B}(\Delta)$ be the space of Bloch functions in the unit disk $\Delta$. For any $f\in \mathcal{B}(\Delta)$, we define the Bloch norm by $$ \|f\|_{\mathcal{B}}=\sup_{|z|<1}|f'(z)|(1-|z|^...
yaoxiao's user avatar
  • 1,706
1 vote
1 answer
132 views

Deriving a specific bound for functions in Hardy Space

Reading some article a while ago I read the following: (here $H^2$ represents the Hardy space) Let $f\in H^2$ be such that $f(0)=1$, and let $0<\lvert\lambda\rvert<1$, then $$\lVert f(\lambda z)\...
Tomas smith Smith's user avatar
1 vote
0 answers
63 views

Extension of meromorphic distribution

Let $W$ be a topological vector space (e.g. Frechet) with a dense subspace $V$. Let $D_s$ be a distribution on $V$ that is meromorphic in $s\in\mathbb C$ and extends continuously to $W$ with respect ...
Tian An's user avatar
  • 3,799
2 votes
1 answer
136 views

Is there a scalar product which makes orthonormal the family of complex functions $ (f_n)_{ n \geq 1 } $?

Let $ (f_n)_{ n \geq 1 } $ be a family of complex functions defined as follow, $ \forall n \geq 1 $, $$ f_n (z) = \dfrac{1}{n^{z}} $$ I would like to ask you if it is possible to construct a ( non-...
Angel65's user avatar
  • 595
2 votes
0 answers
151 views

Does the complex interpolation space $(L^1(\mathbb{R}),W^{2,1}(\mathbb{R}))_{\frac{1}{2}}$ continuously embed into $L^\infty(\mathbb{R})$?

The complex interpolation space between $(L^p(\mathbb{R}),W^{2,p}(\mathbb{R}))_\theta$ with interpolation parameter $\theta=\frac{1}{2}$ is known to be $W^{1,p}(\mathbb{R})$ for $1<p<\infty$. As ...
vmist's user avatar
  • 989
1 vote
1 answer
121 views

An asymptotic integral with complex phase

Suppose that $D\subset \mathbb R^2$ is the closed unit disk and that $f\in C^{\infty}(D)$. Assume that for all $\lambda \in (1,\infty)$ there holds $$ \left|\int_D f(x^1,x^2)\, e^{\lambda (x^1+ix^2)}\,...
Ali's user avatar
  • 4,143
0 votes
1 answer
102 views

On weighted Fourier transforms

Suppose that $f\in L^{\infty}((0,1))$ and that there exists $c_1,c_2>0$ such that $$ \left|\int_0^1 e^{i \xi x} e^{-|\xi|^{-1}x}f(x)\,dx \right| \leq c_1 e^{-c_2|\xi|} \quad \forall\, |\xi|>1.$$ ...
Ali's user avatar
  • 4,143
3 votes
1 answer
249 views

Is Toeplitz operator on the Bergman space bounded iff its symbol is bounded?

It is fairly well known that if $T_\varphi$ is a Toeplitz operator on the Hardy-Hilbert space, then $\lVert T_\varphi \rVert = \lVert \varphi \rVert _{\infty}$. Now, if $\varphi \in L^\infty (\mathbb ...
ash's user avatar
  • 151
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
4 votes
2 answers
360 views

Functions with asymmetrically decreasing Fourier transform?

$\def\ii{{\rm i}}\def\bbR{\mathbb R}\def\bbC{\mathbb C}\def\bbNo{\mathbb N_0}\def\Fou{\mathscr F}$Specifically, I would like to have a compactly supported continuous function $f=u+\ii\,v:\bbR\to\bbC$ ...
TaQ's user avatar
  • 3,584
2 votes
1 answer
112 views

On compactly supported functions with prescribed sparse coordinates

Let $\{\phi_n\}_{n=1}^{\infty}$ be an orthonormal basis for $L^2((0,1))$ consisting of Dirichlet eigenfunctions for the operator $-\partial^2_x + q(x)$ where $q \in C^{\infty}_c((0,1))$ is fixed. ...
Ali's user avatar
  • 4,143
0 votes
1 answer
415 views

Necessary conditions for convergence of convolution

In math.SE, I've asked a question about the convergence of convolution of two functions which have bilateral Laplace transform and also have disjoint Region Of Convergence (ROC) but the question didn'...
S.H.W's user avatar
  • 61
1 vote
0 answers
115 views

Looking for examples of kernels with scalar Pick property but not the complete Pick property

I am studying Pick Interpolation and Hilbert Function Spaces by Agler and McCarthy. A kernel $k$ on a set $X$ is said to have $M_{s,t}$ Pick property whenever $x_1,x_2, \ldots , x_n \in X$ and $W_1, ...
ash's user avatar
  • 151
0 votes
0 answers
94 views

The asymptotic behaviour of the Fourier transform of a certain class of radially symmetric functions

Fix $\theta\in (-\pi/2,\pi/2)$ and let $a>0$. Suppose that $f:\mathbb{C}\rightarrow \mathbb{C}$ is analytic in $S:=\{z\in \mathbb{C}: |\arg{z}|<\pi/2\}$ and $$|f(z)|\sim |z|^{-a},\qquad |z|\to \...
Medo's user avatar
  • 852
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
3 votes
0 answers
111 views

What is known about the analytic continuation of Maz'ya's modified harmonic zeta function $\sum_{n=1}^{\infty} e^{-zH_n}$?

Question: If we let $H_n = \sum_{k=1}^{n} \frac{1}{k}$ be the harmonic numbers then we can consider the modified zeta function $$ f(z) = \sum_{n=1}^{\infty} e^{-zH_n } = \sum_{n=1}^{\infty} e^{-z(\ln(...
Sidharth Ghoshal's user avatar
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
1 vote
1 answer
119 views

weakly separated sequences in RKHS are separated by Gleason metric

I am reading Agler and McCarthy's Pick Interpolation and Hilbert Function spaces. In Chapter 9, the authors ask to observe that weakly separated in a Reproducing kernel hilbert space implies separated ...
ash's user avatar
  • 151
1 vote
0 answers
125 views

Interpolating sequences are strongly separated

I am reading Agler and McCarthy's Pick Interpolation and Hilbert Function spaces. In Chapter 9, titled "Interpolating Sequences", the authors say that every interpolating sequence is ...
ash's user avatar
  • 151
1 vote
1 answer
90 views

The number of roots of pseudo-exponential polynomials

Assume that $J$ is the interval $(-\pi,\pi]$. For $k=1,\ldots,2n$, suppose that $\lambda_k$s are real functions on $J$ with $|\lambda_k|=1$, meaning that $\lambda_k(t)$ is either $-1$ or $1$ where $t\...
ABB's user avatar
  • 4,058
3 votes
0 answers
245 views

Norm on the space of real analytic functions

The space $C^{\omega}(\Omega)$ of real-valued real analytic functions on the open bounded set $\Omega\subset \mathbb R^n$ does not have any obvious or natural metric which would make it a Fréchet ...
Wreck it Ralph's user avatar
1 vote
1 answer
88 views

Mean values of polynomial and holomorphic matrices

Lemma. Assume $H: \mathbb{R} \to \mathbb{R}^{d \times d}$ is a polynomial of degree $m$, such that for all $x \in \mathbb{R}$, $H(x)$ is a symmetric semidefinite matrix. For all $n \geq 0$ and real ...
Sébastien Loisel's user avatar
10 votes
0 answers
652 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
1 vote
0 answers
113 views

Computing a limit for the Weierstrass function

Let $a\in (0,1)$ and let $b$ be an odd positive integer such that $ab>1+\frac{3}{2}\pi$. Let $\alpha \in (0,1)$ be defined by $\alpha= -\frac{ln(a)}{ln(b)}$ and consider the well known Weierstrass ...
Ali's user avatar
  • 4,143
1 vote
0 answers
111 views

Residues of analytic operators

Suppose we have analytic operators $P_{z}: C^1[0,1]\to C^1[0,1]$, where $z \in \mathbb{C}$, and the spectrum of $P_{z_0}$ possesses an isolated eigenvalue $1$ (assuming multiplicity is 1 and $I-P_z$ ...
opera's user avatar
  • 11
4 votes
1 answer
305 views

Holomorphic extension of the Fourier transform of a measure

If an entire holomorphic function $f(z)$ is given by the analytic continuation of $f(x)=\int_\mathbb{R}e^{-ix\xi}\,d\mu(\xi)$ with a finite Borel measure $\mu$ on $\mathbb{R}$, then $g(x):=\int_\...
user509119's user avatar
3 votes
1 answer
166 views

A limit arising from Mellin Inversion: How to compute a specific term of an asymptotic series?

So I am wondering if there exists a general procedure for the following problem: given a monotonically increasing function $f(n)$ which is nonegative on the interval $[0,\infty)$ and grows faster than ...
Sidharth Ghoshal's user avatar
0 votes
0 answers
77 views

Completeness of a normed space

We consider the set $\mathcal{PC}([-r,0],X)$ $$\mathcal{PC}([-r,0],X):=\{\varphi:[-r, 0] \rightarrow X: \varphi \text{ is continuous everywhere except for a finite number of points } t_* \text{ ...
Mathlover's user avatar
4 votes
0 answers
120 views

Matrix product of entire functions

Suppose I have two $d \times d$ entire matrix functions $F, G$ defined on $\mathbb{C}$ with the the property that $\|FG^*\|_{L^\infty(\mathbb{C})} < \infty$. Can anything be said about $F$ and $G$, ...
Joshua Isralowitz's user avatar
2 votes
2 answers
227 views

Hardy space inclusion in the right-half plane

I'm looking for an example of a function $u \in H_2$ such that $u \notin H_\infty$, where $H_p$ is the Hardy space on the right-half plane. Since this notation is perhaps not standard, here is a ...
Laurent Lessard's user avatar
5 votes
1 answer
358 views

Is there a meaningful interpretation of an $L^i$-space?

Do complex-normed spaces exist? Is there an extension of $p$-norms to $p\in\Bbb C\setminus\Bbb R$? A while ago I thought of extending $L^p$-spaces to the complex-normed setting. After some discussions,...
TheSimpliFire's user avatar
4 votes
0 answers
74 views

Is there a dense set of Lipschitz functions in $H^\infty(U)$, each of which maps $(1,0,\ldots,0)$ to 1, where $U$ is the unit ball in $\mathbb{C}^N$?

Let $U$ be the open unit ball in $\mathbb{C}^N$, let $A(U)$ be the algebra of functions analytic on $U$ and continuous on $\bar U$, and let $u=(1,0,\ldots,0)$. Let $\mathcal{B}=\{f\in H^\infty (U): \|...
David Walmsley's user avatar
3 votes
1 answer
190 views

Can the equation $1+z+z^q=z^n$ have multiple complex roots $z$?

It is proved here that the equation $1+z+z^2=z^n$ have no multiple complex roots. Q. Let us consider the equation $1+z+z^q=z^n$ where $q$ and $n$ are natural numbers with $1<q<n$. Any ...
ABB's user avatar
  • 4,058
1 vote
0 answers
140 views

Does a Borel transform uniquely determine a Borel measure?

It is a known fact that Borel measures are uniquely determined by their Fourier transforms. This is the motivation for the following question. I came across the concept of a Borel transform of a Borel ...
JustWannaKnow's user avatar

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