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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Exercise on Hardy spaces

This exercise on Hardy spaces looks fairly simple, but I don't know how to tackle it. Any hints or solutions would be appreciated: Let $g \in \mathcal{H}(\mathbb{D}).$ For $0<p<+\infty$ and $...
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30 views

Dominated convergence theorem for $\lim_{r\to 1^-} \int_0^{2\pi} \text{Log} \left|1-re^{it}\right| \, dt$

I am trying to apply the DCT to see that $$\lim_{r\to 1^-} \int_0^{2\pi} \text{Log} \left|1-re^{it}\right| \, dt $$ exists and is finite, where $\text{Log}$ is the main branch of the logarithm. It is ...
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44 views

$\text{Log}(1-z) \in H^p(\mathbb{D})$ for all $0<p<\infty$

Apparently, it is well known that $\text{Log}(1-z) \in H^p(\mathbb{D})$ for all $0<p<\infty,$ where $\text{Log}$ denotes the principal complex logarithm. How can this be proven?
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Questions while reading “$H^p$ spaces of several variables(Fefferman and Stein)”.

I have two questions while reading this paper: Hp spaces of several variables." Acta math 129 (1972): 167-193. Question 1. Line $7$ on page $147$ to line $13$ on page $148$ is a proof of "Theorem 3$\...
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1answer
38 views

Criteria for function to be of Hardy class $H^p$

I was reading some papers, and I don't know how to prove following statement: Let $f$ be conformal map of $\mathbb{D}$ onto an unbounded domain,such that, $f(0)=0$ and $dA$ denote Lebesgue measure on ...
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28 views

Hardy space on the upper half plane and entire functions

Consider a function belonging to the Hardy space on the upper half-plane (for definiteness, let's focus on square-integrable functions). Is there any theorem stating under which conditions is such a ...
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1answer
25 views

Let $f\in L^{\infty}$, its that true $Pf\in H^{\infty}$?

My question is: Let $f\in L^{\infty}[S^1]\subseteq L^2[S^1]$, is that always true for $Pf\in H^{\infty}[S^1]\subseteq H^2[S^1]$? Here, $S^1$ is the unit circle in complex plane, and $H^{\infty}$ ...
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72 views

need to get rid of a term inside a determinant

I think it is not possible, but that is with my limited mathematical knowledge. Perhaps someone knows a determinant/matrix identity that I have not come across yet. Consider a block matrix P and V ...
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59 views

Functions on the unit circle having finitely many zeroes

I actually consider functions on the strip $S(0,\pi) = \{ z | 0 < Im(z)<\pi \}$ in the complex plane, but by a biholomorphism, I can map to the unit disc. In particular, the functions I consider ...
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30 views

Question about analytic functions

Let us consider $H^\infty(\mathbb{D}, X)$ the space of bounded analytic functions on the unit disk with values in a Banach space $X$, and $A(\mathbb{D},X)$ the closure of analytic polynomials in the ...
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Weighted Hardy space composition

Let $H^2(\omega)$ be the closure of the set of analytic on the unit circle $T$ polynomials, with respect to the scalar product $$ \langle f, g\rangle_\omega = \oint_T f \bar{g} \omega \,\mathrm{d}z, $$...
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Is $H^\infty(\mathbb{C_+}) \subset H^2(\mathbb{C_+})$?

I know that $H^\infty(\mathcal{D}(0,1)) \subset H^2(\mathcal{D}(0,1))$, and that there is a mapping $$f: \mathcal{D}(0,1)\to \mathbb{C}_+ : z\mapsto s=\frac{1-z}{1+z}.$$ However in $\mathbb{C_+}$ I ...
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66 views

Hardy inner product for the right half plane

The Hardy space $H^\infty(\mathbb{C_+})$ is defined to be the space of all functions $f$ >analytic and bounded on the complex right half plane $\mathbb{C_+}$ with the norm $\lVert \cdot \rVert_H$ ...
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1answer
65 views

Question on Douglas, Banach Algebra, Hardy spaces

In the text by Douglas, Banach Algebra techniques in Operator Theory, pg. 25, 1.47 Hardy spaces was introduced. If $\mathbb {T}$ denotes the unit circle in the complex plane, let $\chi_n $ denote ...
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82 views

Wiener's tauberian theorem for Hardy space

For $a>0$ let us define $$H^2(-a,a)=\{f \ \mbox{is analytic in the strip $|\Im(z)|<a$}: \sup_{y\in [-a,a]}\int_{\mathbb{R}}|f(x+iy)|^2\,dx<\infty\}.$$ For $f\in H^2(-a,a)$, define $\|f\|=\...
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18 views

How much can the hypotheses of the Schwarz integral formula be weakened?

According to https://en.wikipedia.org/wiki/Schwarz_integral_formula, which I will paraphrase here, if $f$ is a holomorphic on the closed unit disk $\{ \lvert z \rvert \leq 1 \}$, then $f(z) = \frac{1}...
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82 views

Is a de-zeroed Hardy function still Hardy?

Suppose a Hardy function $f(z)$ on the upper half complex plane or $f\in H^{2+}$ (Chapter II, p.45 of Fulvio Ricci, Hardy Spaces in One Complex Variable) has a zero of order $m$ at $\omega$ with $\...
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What is the closure of $h^2(U)$ in the topology of the uniform convergence on upper half planes?

Let $U:=\{z\in\mathbb{C}\ |\ \operatorname{Im}(z)>0\}$. Let $h(U,\mathbb{R})$ be the set of real harmonic functions defined on $U$. Define $$h_0(U,\mathbb{R}):=\left\{u\in h(U,\mathbb{R})\ |\ \left(...
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36 views

If $f$ is in a weighted Bergman space for the upper half plane, then $\forall\varepsilon>0, z\mapsto f(z+i\varepsilon)$ is in the Hardy space.

Let $U:=\{z\in\mathbb{C}\ |\ \operatorname{Im}(z)>0\}$ and denote with $H(U)$ the set of holomorphic functions on $U$. Define: $$H^2(U):=\{f\in H(U) \ |\ \sup_{y>0}\int_\mathbb{R} |f(x+iy)|^2\...
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64 views

Looking for an explicit $f\in C(\mathbb{T})$ such that $t\mapsto\sum_{n=0}^{\infty}\hat{f}(n)e^{int}\notin L^\infty(\mathbb{T})$.

Denote the 1-torus with $\mathbb{T}$ and if $f\in L^1(\mathbb{T})$ denote the Fourier transform of $f$ by $\hat{f}$. I know that $$\exists f\in C(\mathbb{T}), t\mapsto\sum_{n=0}^{\infty}\hat{f}(n)e^{...
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124 views

Convergence of Fourier series in Bergman norm

Let $D$ be the unit disk, let $\nu>-1$, let $\varphi_\nu(w):=(1-|w|^2)^\nu$, let $\operatorname{d}\mu_\nu:=\varphi_\nu\operatorname{d}\mu$ where $\mu$ is the Lebesgue measure on the unit disk. If $...
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55 views

Range of the Bergman projection.

If $D$ is the unit disk in the complex plane, $\mu$ is the Lebesgue measure on $D$ and $H(D)$ is the space of holomorphic functions on $D$, define the Bergman projection by: $$B:L^1(\mu)\rightarrow H(...
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140 views

Regarding Poisson kernel being in $h^p(\mathbb{D})$

Let $\mathbb{D}$ denote the open unit disc in the complex plane. Let $h^p(\mathbb{D})$ be the space of all harmonic functions $f$ on $\mathbb{D}$ such that $$\sup_{0\leq r<1}\left(\frac{1}{2\pi}\...
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1answer
56 views

The spectrum of the Hardy Banach algebra $(H^1(\mathbb{T}),+,*,\|\|_1)$.

Let $\mathbb{T}$ be the $1$-torus and define: $$H^1(\mathbb{T}):=\{f\in L^1(\mathbb{T})\ | \ \forall n<0, \hat{f}(n)=0\},$$ where if $f\in L^1(\mathbb{T})$ we have denoted by $\hat{f}$ the Fourier ...
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1answer
205 views

Does a holomorphic function on the unit disk with continuous radial limits have a continuous extension to the closed disk?

Question: does there exist a holomorphic function $f$ defined on the unit disk $D$ such that $\forall t \in \mathbb{R}, \exists \lim _{r\rightarrow 1^-} f(re^{it})\in\mathbb{C}$; the periodic ...
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1answer
102 views

Cauchy Integral theorem and Hardy Spaces

I am working through Theory of $H^p$ Spaces by Peter L. Duren, and I am having some difficulty with proof of the following theorem: Theorem. Every function $f \in H^1$ can be expressed as the ...
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191 views

Is harmonic Hardy space $h^p (D)$ complete for $0<p<1$?

Let $D$ be the unit disk and $0<p<1$. Let $h^p(D)$ be the harmonic Hardy space on the unit disk (i.e. $u\in h^p(D)$ if and only if $u$ is harmonic on the disk and $\sup_{0<r<1}\int_{-\pi}^\...
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87 views

A (counter)example in harmonic Hardy spaces.

I'm looking for an example of an harmonic function in the unit disk $D$, say $u$, such that the family $$(t\mapsto u(re^{it}))_{r\in[0,1)}$$ is uniformly integrable on the torus $\mathbb{T}$ and such ...
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195 views

Convergence of Taylor series in Hardy norm

If $0< p<\infty$, let $H^p(D)$ denote the Hardy space on the unit disk. We know from Hardy spaces theory that, for every $f\in H^p(D)$, the family of functions $$f_r :D\rightarrow \mathbb{C}, z\...
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1answer
95 views

Norm convergence for Fourier series in Hardy spaces with $0<p<1$

Fix $0<p<1$. Let $D$ be the complex unit disk and let $\mathbb{T}$ be the torus. Let ${\frak{H}}^p(\mathbb{T})$ denotes the $L^p(\mathbb{T})$ closure of the holomorphic trigonometric polynomials,...
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1answer
276 views

Analytic function on the right half-plane, continuous and bounded on the closed right-half plane

Let $\mathbb{C}_+$ be the right half-plane defined by \begin{align} \mathbb{C}_+ \stackrel{\text{def}}{=} \{z \in \mathbb{C} \ \colon \Re(z) > 0\}. \end{align} I am trying to understand why for ...
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50 views

Regarding invertible functions being outer

I am reading these notes titled Hardy spaces, inner and outer functions, shift-invariant subspaces, Toeplitz and Hankel operators In the second last paragraph, it says that any function that is ...
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182 views

Are polynomials dense in $H^\infty$

Let $H^\infty(D)=\{f:D\longrightarrow \mathbb{C}: f \;\text{is bounded and analytic on}\; {D}\}$ where $D=\{z\in\mathbb{C}: |z|<1\}$. Are the polynomials dense in $H^\infty(D)$? I know that they ...
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150 views

Inner and outer factors in Hardy spaces

Let $f(z) = \sin(z),$ viewed as a function in the hardy space $H^1.$ How do I factor $f$ into inner and outer factors? I know that the formula for the outer factor is $$Q_f(z)= \exp(\dfrac{1}{2\pi} \...
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242 views

A proof for a Hardy inequality for holomorphic functions in Hardy space $H^1$

Let $H^1$ be the Hardy space of holomorphic functions on the unit disk. Where can I find a proof (as simple and selfcontained as possible) of this result? If $f\in H^1$, then $$\sum_{n=0}^{\infty}\...
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1answer
104 views

Characterization of a positive finite Borel measure on the circle

Let $\mu$ be a positive finite measure on the unit circle $\mathbb{T}$. Then the set $span\{z^n:n\in \mathbb{Z}\}$ is dense in $L^2(\mu)$. Let $H^2(\mu)$ be the closure ( in $L^2(\mu)$ ) of the set $...
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44 views

Can one take the Hilbert transform of an infinite-time “filtered” signal?

This question essentially asks whether the associativity of convolutions can be extended to cover the Hilbert transform, even when one does not know a priori that the Hilbert transform can be applied. ...
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If $\varphi$ is a non-constant inner function in $\mathbb{D}$, then $1/\varphi \notin H^p$ for $p>0$

This is a problem from Rudin's Real and complex analysis. If $\varphi$ is a non-constant, non zero inner function in $\mathbb{D}$, then $1/\varphi \notin H^p$ for $p>0$. Inner function is a ...
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1answer
129 views

Understanding Elements of Hardy Spaces

As I was reading HARDY SPACES $H^p$, It's definition clearly states that it is the space of analytic functions on a unit disc satisfying norm conditions. I didn't get that ,the texts say that $H^p$ ...
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125 views

About strict inclusion in Hardy spaces

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 ...
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254 views

If a function belongs to the Hardy space

Let $\mathbb D= \{z\in\mathbb{C}: |z|<1\}$. Let $\operatorname{Hol}(\mathbb {D})$ denote the space of holomorphic functions on $\mathbb D$. The Hardy spaces on $\mathbb D$ are defined as follows. ...
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1answer
145 views

Original paper of F. &M. Riesz the theorem

This is not actually a question! I need a copy of F. & M. Riesz theorem F. and M. Riesz, Über die Randwerte einer analytischen Funktion, Quatrième Congrès des Mathématiciens Scandinaves, ...
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124 views

The Hilbert space of Dirichlet Series, with square-summable coefficients is not an algebra

Consider the Hilbert space of Dirichlet series $$\mathcal{H}^2=\lbrace f(s)=\sum_{n=1}^{\infty}\dfrac{a_n}{n^s}:\sum_{n=1}^{\infty}|a_n|^2<\infty\rbrace,$$ with norm $\|f\|_2^2=\sum_{n=1}^{\infty}|...
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105 views

Commutant of unilateral shift

Let $H^2$ be the Hardy space and let $A$ be a bounded operator on $H^2$ such that Lat $U\subseteq$ Lat $A$ where $U$ is the unilateral shift. Show that $AU=UA$. (Here Lat $A$ denotes the set of all ...
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1answer
84 views

Greatest common divisors of bounded holomorphic functions

Let $\mathbb{D}$ be the open unit disk and let $H^\infty (\mathbb{D})$ be the Banach algebra of bounded holomorphic functions. Let $f_1,\ldots , f_n\in H^\infty (\mathbb{D})$. Now there is a greatest ...
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483 views

Hardy Spaces: Meaning of $\frac{1}{z}\mathcal{RH}_\infty$

Question I'm reading a paper (p.4) on linear systems where the authors silently introduce the notation for the set $\frac{1}{z}\mathcal{RH}_\infty$. Unfortunately I don't get the meaning of the ...
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1answer
100 views

Model space from Blaschke factor

Let $u$ be an inner function and denote by $H^2$ the Hardy space on the open unit disc $\mathbb{D}$. A model space $K_u$ associated to $u$ is a Hilbert space of the form \begin{equation} K_u = (...
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1answer
131 views

Hardy space as a Banach lattice

The Hardy spaces $H^p$ of holomorphic functions on the unit disk are Banach spaces. Question: Are they also Banach lattices? If yes, why is it less common to consider the Hardy spaces as Banach ...
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2answers
162 views

If a function belongs to $H^p$

Let $\mathbb D$ denote the open unit disc in $\mathbb{C}$ . Let $Hol(\mathbb {D})$ denote the space of holomorphic functions on $\mathbb D$. The Hardy spaces on $\mathbb D$ are defined as follows. $$...
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1answer
79 views

How to determine $p$ for which $\frac{1}{1-z} \in H^p(\mathbb{D})$

I want to figure out for which $p \in (0,\infty)$ it is true that $\frac{1}{1-z} \in H^p(\mathbb{D})$, where $H^p(\mathbb{D})$ is the space of analytic functions $f$ on $\mathbb{D}$ such that $\sup_{0 ...