Questions tagged [hardy-spaces]

For questions about Hardy spaces. Use the other related tag like (tag: complex-analysis) or (operator-theory).

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Bounding $f'(z)$ with $O(\log(\frac{1}{1-r}))$ for an Analytic Series

I am working with an analytic function defined within the unit disk $|z| < 1$ as follows: $$ f(z) = \sum_{n=1}^{\infty} a_{n} z^{n}, $$ where I have the condition that $\sum_{n=1}^{N} n|a_{n}| = O(\...
El Sh's user avatar
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Boundedness of derivative related to hardy spaces

Assume $f:[0,1]\to [0,1]$ is an orientation-preserving homeomorphism, and $f$ is absolutely continuous on $[0,1]$, $\log f'\in H^{1/2}([0,1])$, that is to say, $$\int_{0}^{1}\int_{0}^{1}\frac{|\log f'(...
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About the algebra $H^\infty + C(\mathbb{T})$

We know that $H^\infty + C(\mathbb{T})$ is the closed subalgebra of $L^\infty(\mathbb{T})$ containing $H^\infty$. How to show that $H^\infty + C(\mathbb{T})$ = clos$[\cup_{n\geq 0} \chi_{-n} H^\infty$]...
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Fejer-Riesz Theorem for polynomials

Let $\mathbb{D}=\{z: |z| <1\}$ and $\mathbb{T}=\{z: |z|=1\}$. Suppose $D$ and $E$ are polynomials of degree atmost $n$ with complex coefficients such that $|E(z)| \leq |D(z)|$ for all $z \in \...
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Why is this assumption necessary on the proof of duality of $H^1$ and BMO? (Stein)

I was going through the proof on Stein's Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals of the fact that BMO is the dual of $H^1$ (this is on Chapter IV, section 1.2....
confusedTurtle's user avatar
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Why is von Neumann inequality important for equivalence of $\forall_j \ T_j^n\rightarrow 0$ in A-topology and abs continuity of $(T_1,\ldots, T_N)$?

The whole theorem goes as follows: Let $(T_1, \ldots, T_N)$ be a tuple of commuting operators in Hilbert space $H$ satisfying: $$\exists_{M > 0} \ : \ \forall_{p \in \mathbb{C}[z_1, \ldots, z_N]} \ ...
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Closure of $H^{\infty}$ with BMO-norm

Let $H^{\infty}$ denote the space of bounded analytic functions on the unit disk $\mathbb{D}$. Let $BMOA$ be the subspace of Hardy Space $H^2$, such that $$\|f\|_{BMOA}=|f(0)|+\sup_{a\in\mathbb{D}}\|f\...
SprtWhitebeard's user avatar
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Is there an "easy" proof of the existence of radial limits of functions in the hardy spaces $H^p$?

Let $HOL(\mathbb{D})$ be the analytic functions defined on the unit disk and for $1\leq p \leq \infty$, $$H^p = \{f\in HOL(\mathbb{D}) : \lim_{r\nearrow 1} \int_{rC}|f(z)|^p \frac{dz}{2\pi i} < \...
travelingbones's user avatar
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Boundedness of functions in the Hardy space $H^2(\mathbb H)$

Let $F: \mathbb H \to \mathbb C$ be a function in the Hardy space $H^2(\mathbb H)$. In other words, we have $$\sup_{y>0} \int_{-\infty}^\infty |F(x+iy)|^2 \, dx < \infty.$$ Let $f(x) = \lim_{y \...
Laplacian's user avatar
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Does these conditions ensure Hardy space membership?

Let D be a disk of center $c$ and radius $R>0$ located in the complex plane, $\partial D$ be the boundary of the disk, $H(D)$ denote the collection of analytic functions defined on $D$, $$H^p(D)=\...
Mathitis's user avatar
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Example of a function which is in Hardy space $H^1(\mathbb{D})$ but not in $H^2(\mathbb{D})$

We know that the Hardy spaces on unit disc $\mathbb{D}$, $H^2(\mathbb{D})\subset H^1(\mathbb{D})$. I need to find an example to show that the containment is proper. I was trying to use the fact that ...
Meow's user avatar
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Find a weighted Sobolev space $X\subset L^1(\mathbb R^n)$ such that, if $u\in X$ and has zero mean, then $u$ is in the Hardy space $\mathcal H^1$.

I have asked nearly the same question on Math Overflow, but it is maybe too low level. Let $\mathcal H^1(\mathbb R^n)$ be the real Hardy space (as in Stein's book, Chapter 3). It is well known that $\...
Lorenzo Pompili's user avatar
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Orthogonal complement of $BH^2$, where B is Blaschke

It is known that if $B=\prod_{i=1}^{\infty} \frac{z-z_i}{1-\overline{z_i}z}$ where $z_1, z_2,...$ are all distinct and $\sum_{i=1}^{\infty}(1-|z_n|)<\infty$, then $(BH^2)^{\perp}=\overline{span}\{\...
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Riesz representation theorem for functionals acting on Hölder $C^\alpha$ functions

Assume you have a linear functional $F:C^\alpha(\mathbb R^n) \mapsto\mathbb R$ such that $$ |F(f)| \leq \vert f \vert_{C^\alpha(\mathbb R^n)} $$ but only depending on the Holder seminorm, that is, $$ ...
HHN's user avatar
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Vanishing of one-half of Fourier coefficients implies boundary of $H^p$-space function

Let $p\in [1, \infty]$. Suppose $g \in L^p(T)$ where $T$ is the unit circle. I want to prove g is boundary value of $f \in H^p$ if and only if $\int_{-\pi}^{\pi}g(e^{it})e^{-int} dt = 0$ for all $n \...
Jiya's user avatar
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Functional proof of nonexistence of an integral pairing

It is well known that $(H^1)^*=BMO$ (mod some identifications). The way this is usually proved is by proving that every continuous linear functional on a suitable dense subspace $X$ of $H^1$ can be ...
Pelota's user avatar
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restrictions of analytic functions to the circle

This seems really simple, but since I didn't find it written anywhere I wanted to make sure I'm not crazy and/or get a reference to a standard textbook where this fact appears. Let $f:\mathbb{C}\to\...
Amir Sagiv's user avatar
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Every simple functions in $L_p(\mathbb{T}) $ is a trigonometric polynomial.

I know for a finite measure space $(\Omega, \mathcal{A}, \mu)$, Simple functions are dense in $L_p(\mu)$. Also it is true that, if we consider normalized Lebesgue measure on the unit circle then ...
Smita's user avatar
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35 votes
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Major error in classic "Banach Spaces of Analytic Functions"

"Banach Spaces of Analytic Functions" by Kenneth Hoffman is an excellent introduction to $H^{p}$ spaces and is considered a classic mathematical analysis textbook. I was therefore surprised ...
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1 answer
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Checking that a function is in Hardy-2 space of the unit disc around the origin

I want to check that the function $f(s) = \frac{1}{(s-1)^{0.3}}$ is in $H^2(\mathcal{D(0,1)})$ which is a vector space of holomorphic functions that satisfy $$ \sup _{0 \leqslant r<1}\left(\frac{1}{...
Yakov's user avatar
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Hardy operator and its adjoint

For nonnegative function $f$ on $\mathbb R^n$, $$Hf(x)=\frac{1}{\Omega_n \left|x\right|^n} \int_{\left|y\right|<\left|x\right|}f(y)dy, \quad H^*f(x)= \int_{\left|y\right|\geq\left|x\right|}\frac{f(...
Luis De Oro's user avatar
2 votes
1 answer
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Functions in $A(\mathbb{C})$ that vanish at zero [closed]

Let $A\subset L^{\infty}(-\pi,\pi)$ be the closed subspace spanned by $\{e^{inx}\}_{n\geq0}$, i.e., all continuous functions on the unit circle that are uniform limits of trigonometric polynomials of ...
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Blaschke product: What does extending continuously mean here?

I am studying Factorisation of $H^p$ functions from Hoffman's Banach Spaces of Analytic Functions. The author is talking about extending the Blaschke product continuously to the accumulation points of ...
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Rudin RCA Chapter 17 Exercise 3

I am going through Rudin's RCA (3rd Ed), and it seems to me that Chapter 17 ($H^p$-Spaces), Exercise 3 (the initial part) has an incorrect equivalency statement. Namely, it states the following: ...
Hassaan Naeem's user avatar
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1 answer
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Blaschke product having poles proof

I am studying Banach Spaces of Analytic Function by Hoffman. Hoffman proves the following theorem: The Blaschke product whose zeroes are \begin{align*} \alpha_{1} , \alpha_2 , \ldots \end{align*} ...
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Is every holomorphic $L^p(\mathbb{D})$ function in $H^p$?

Let $f\in H^p$ where $H^p$ denotes the Hardy space for $1\leq p<\infty$. That is, $f$ is a holomorphic function on the open unit disk, $\mathbb{D}$, such that $$\sup_{0<r<1}\int_0^{2\pi}\vert ...
Anon's user avatar
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hardy spaces: f is analytic in the closed disc implies the singular part of f is zero

I am studying Banach Spaces of Analytic Function by Hoffman. After giving a description of nonzero $H^1$ functions as a factorization of Blaschke, Singular and Outer function, the author makes a ...
ashK's user avatar
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Cauchy transform in polydisc

It is well known that for $p\in(1,\infty)$ and any $f\in L^p(\mathbb{T})$ function $Cf(z)=\int\limits_{\mathbb{T}}\frac{f(\zeta)}{1-\bar\zeta z}dm(\zeta)$ (where $m$ is normalized Lebesgue measure on ...
Timur B.'s user avatar
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3 votes
1 answer
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Hoffman - Definition of Blaschke product

I am self-studying Banach Spaces of Analytic Functions by Hoffman. I have a question regarding the following theorem: According to the author, the set $K$ formed as in the theorem is compact. I do ...
ashK's user avatar
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Hardy Spaces - outer function is holomorphic

I am studying Banach Spaces of Analytic Functions by Hoffman. In Chapter 5 Page 61, the textbook claims that If $u \in L^1 (\mathbb T)$ then the function $F: \mathbb D \to \mathbb C$ \begin{align*} F(...
ashK's user avatar
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Hardy Spaces - Proving that the norm is well defined

I am self studying Banach Spaces of Analytic Function by Hoffman. In the Chapter 3 titled "Analytic and Harmonic Functions in the Unit Disc", the author defines the class $H^p$ for $1\le p \...
ashK's user avatar
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2 votes
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Hardy class of bounded analytic functions is Banach space

I need help with the following: Let $H^{\infty}(\mathbb{D})$ denote Hardy class of bounded analytic functions on unit disc $\mathbb{D} = \{z \in \mathbb{C}: |z|<1\}$. Prove that $$||f|| = \...
Maria's user avatar
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0 answers
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Prove that, $H^\infty+C(\Bbb{T})$ is not a $C^*$-algebra

Here $H^\infty:=\left\{f\in L^\infty(\Bbb{T}):\ \frac{1}{2\pi}\int\limits_0^{2\pi} f\chi_{-n}\ dt=0\ \forall n<0\right\}$ i.e. all those $L^\infty$ functions whose negative fourier coefficients are ...
MathBS's user avatar
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Show $\lim_{t \to 0} \frac{\sigma\left( \left\{ e^{is} : s \in (\theta - t, \theta + t \right\} \right)}{2t}=0$ for singular measure $\sigma$ of $T$

I am self studying "Representation Theorems in Hardy Spaces" by Javad Mashreghi. Here's a claim [in Page 72 of the textbook] that I am unable to prove: If $\sigma$ is a singular measure of $\...
ashK's user avatar
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2 votes
1 answer
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Inverting series with logs and W

You've all heard it: what does a drowning analytic number theorist say? Log log log log.... I very frequently deal with the sorts of functions that one comes across and want to invert them. Generally ...
Charles's user avatar
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If sequence of finite Blaschke Products $(B_n)_{n\in N}$ converges uniformly to $B$ then $\frac{B_n(z)}{z^m}$ converges uniformly to $\frac{B}{z^m}$

I am having problem with showing that if sequence of finite Blaschke Products $$B_n=z^m\prod_{k=1}^{n}\frac{|z_k|}{z_k}\frac{z_k - z}{1 - \overline{z}_k z}$$ where $(z_n)_{n\in \mathbb{N}} \subset \...
gengus's user avatar
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2 votes
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Approximations of BMO functions

I couldn't think of a good title. There was a small detail when I was trying to prove something about the norm of BMO functions. This small detail is not directly related to the problem itself, but ...
eggplant's user avatar
2 votes
1 answer
82 views

Let $f\in L^2(\Bbb{T}),\ f\ne0$ and $M=\overline{f\wp_+}$. Prove that, $M$ is simply invariant for $\chi_1$ iff $f\notin \chi_N M$ for some $N>0$

Here we denote $\chi_n$ to be the functions on $\Bbb{T}$ defined as $\chi_n(z)=z^n\ \forall z\in\Bbb{T}$ for $n\in\Bbb{Z}$. We define $\wp_+=\{\sum\limits_{n=0}^N a_n\chi_n|\ a_n\in\Bbb{C},N\ge0\}$ be ...
MathBS's user avatar
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$\int_{-\pi}^{\pi}{\frac{d\theta}{[(1-r)^2 + 2\pi^{-2}\theta^2]^p}} < \frac{1}{(1-r)^{2p-1}}\int_{-\infty}^{\infty}{\frac{dt}{[1+2\pi^{-2}t^2]^p}}$

I found this inequality from Duren's book Theory of Hp-spaces(2000) page 66. According to the book inequality should be true for $\frac{1}{2}\leq r < 1$ and $p>\frac{1}{2}$. I have tried to ...
gengus's user avatar
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What is the domain of functions in $\mathcal H_2$?

In a section of a book that I'm reading about Hardy spaces, the author writes $\mathcal H_2$ is a (closed) subspace of $\mathcal L_2(j \mathbb R)$ with matrix functions $F(s)$ analytic in $\text{Re}(...
mhdadk's user avatar
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$H^p$ is not a Banach space for $p<1$?

As in E. Steins's "Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals" consider the space $H^p$ (with $p < 1$), which contains all tempered distributions $f ...
bs_math's user avatar
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0 votes
1 answer
147 views

Are Hardy spaces Banach algebras?

Let $\mathbb D= \{z\in\mathbb{C}: |z|<1\}$. Let $Hol(\mathbb {D})$ denote the space of holomorphic functions on $\mathbb D$. The Hardy spaces on $\mathbb D$ are defined as follows. $$H^p=\{f\in ...
user531706's user avatar
1 vote
0 answers
54 views

What's the BMO norm of the sine function?

In other words, what is the value of the maximum mean oscillation of the sine function over an interval, \begin{align} || \sin ||_{\mathrm{BMO}} &= \sup_{a,b\in\mathbb{R}} \frac{1}{|b-a|}\int_a^b ...
ABC's user avatar
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1 vote
0 answers
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The Mellin transform as a mapping from a Hardy space to a weighted space

As far as I know, the following fact must be published somewhere, and I would like to find a reference. The Mellin transform is defined by $f\mapsto\frac1{\sqrt{2\pi}}\int_0^\infty f(x)x^{s-1}\,dx$. ...
Limanac's user avatar
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0 answers
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Absolute value of functions in $H^1(\mathbb R^n)$

Let $f$ be a function in the atomic Hardy space $H^1_{at}(\mathbb R^n)$. That is, there exists a sequence of atoms $a_j$ satisfying supp $a_j \subset B_j$ for some ball $B_j$, $\int a_j dx = 0$, $\...
HHN's user avatar
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1 vote
1 answer
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$H^2$ does not contain any rational functions with poles on the unit circle

The Hardy-hilbert space, $H^2$, consists of all analytic functions having power series representations with square-summable complex coefficients. That is, $$H^2=\{f : f(z)=\sum_{n=0}^{\infty} a_n z^ns....
Théo Pontasse's user avatar
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0 answers
129 views

Why does the Laplace transform of a function in L2 belong in the Hardy space?

Does anyone have a proof / mathematical explanation to why the Laplace transform of a function of $\mathbb{L}^2(0,\infty )$ belongs to the Hardy space? Any guidance would be fantastic! I cant find ...
John Bresnan's user avatar
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60 views

a generalized Jensen's equality for Nevanlinna class

Define the Nevanlinna class of holomorphic functions on unit disk by $\displaystyle N(\mathbb{D})=\left\{f\in\text{Hol}(\mathbb{D}):\sup_{0<r<1}\int_\mathbb{T}\log^+|f_r|dm<\infty\right\}$ ...
am_11235...'s user avatar
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5 votes
0 answers
101 views

Showing $\arcsin(z)-i\operatorname{Log}(-iz)-i\log 2 \in H^2(\mathbb{H})$

In order to prove a result about Hilbert transforms, I need to show the complex function $F(z) = \arcsin(z)-i\operatorname{Log}(-iz)-i\log 2$ lies in $H^2(\mathbb{H})$, the Hardy Space for the upper ...
chessman's user avatar
3 votes
1 answer
90 views

containment of $H^1$ spaces into $L^1$ spaces

Let $\mathbb{D}$ denote the unit disk and $\mathbb{T}$ the unit circle , if $f\in H^1(\mathbb{D})$ is outer , then necessarily $\log|f|\in L^1(\mathbb{T})$ ? We know from theory of Hardy spaces that ...
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