# Questions tagged [hardy-spaces]

The hardy-spaces tag has no usage guidance.

20
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### Special function in the Hardy space

Let $H^2(\mathbb{D})$ denote the complex Hardy space, this is: analytic functions defined unit disc $\mathbb{D}$ whose coefficients form a sequence in $\ell^2$. Functions in $H^2(\mathbb{D})$ have a ...

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+100

### Inclusion in Hardy-Smirnov spaces for the analytic continuation of a Cauchy-Type integral with a continuous boundary function

Let $D$ be a bounded simply connected domain in the complex plane $\mathbb{C}=\{z=x+iy\}$ with a Jordan rectifiable boundary $C=\partial D$. Let $P_1$ and $P_2$ be two distinct points on $C$, and let ...

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### 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)\...

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### Function samplable from the past and Hardy spaces

What I am ultimately looking for is a $L^2$ function $f$ on the real line that can be sampled from the past, i.e. for each $x<0$ there are $L^2$ coefficients $c_n(x)$, $n\in \mathbb{N}$ such that, ...

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### $H^1$ and $BMO$ interpoaltion on spaces of homogeneous type

In this Coifman and Weiss I've found that in spaces of homoegeneous type holds a $(H^1,L^1)-(L^p,L^p)$ interpolation theorem. (it is even stronger because it only needs weak $(H^1,1)$ and weak $(p,p)$ ...

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### Completeness of the atomic Hardy space $H^{p,q}(X)$ on spaces of homogeneous type

I have a problem with the following fact: the article by Coifman and Weiss, "Extensions of Hardy spaces and their use in analysis" https://projecteuclid.org/journals/bulletin-of-the-american-...

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### $H^\infty$ functions with certain $H^2$ factors

While discussing the factorization theorems and shift-cyclicity in Hardy spaces, a friend and I came across a problem that seems to be answerable but we could not get anywhere. The problem is as ...

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### Example of an $H^1$ function on the bidisk that is not a product of two $H^2$ functions

Fix $n \in \mathbb{N}$ and consider the Hardy space $H^1 := H^1(\mathbb{D}^n)$, consisting of holomorphic functions $f$ on the unit polydisk $\mathbb{D}^n=\mathbb{D}\times\dots\times\mathbb{D}$ such ...

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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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### Criteria for Hardy space membership

Assume that $p>2$ and let $H^p$ be the Hardy space of holomorphic functions in the unit disk $D$. It seems that $f\in H^p$ if and only if $$P(f):=\int_0^{2\pi}\left(\int_0^{1}|f'(re^{it})|^2(1-r)dr\...

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129
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### An inner product and projection property in RKHS

I'm reading an article regarding the condition for the existence of a solution to a constrained Pick interpolation problem, and the author wrote the following equality - for background, we have the ...

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101
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### Sum power series not continuous unit circle

This is (probably) not a research question and I already asked it on StackExchange but I got no answer over there.
Let us consider the sequence $(a_n)_{\geq 1} = \left(\frac{\cos(2\sqrt{2n}+\frac{\pi}{...

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### $BMO$ is not reflexive

It is well known that $BMO$ is the dual space of the Hardy space $H^1$, which is the dual space of $VMO$. I believe that $BMO$ is not reflexive, but I am not quite sure that the above information is ...

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### Localize functions in the Hardy space $\mathcal H^1(\mathbb R^n)$

Let $f$ belong to the Hardy space $\mathcal H^1(\mathbb R^n)$, $B\subset \mathbb R^n$ be the unit ball. Does there exist a $\bar f\in \mathcal H^1(\mathbb R^n)$ with compact support such that $\bar f=...

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### Existence of the special entire Hardy space function with infinitely many zeros in the strip

Question. Does there exist an entire function $h$ satisfying three following assertions:
$h$ belongs to the $H^2$ Hardy space in every horizontal upper half-plane;
$zh - 1$ belongs to $H^2(\mathbb{C}...

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### Existence of nonzero entire function with restrictions of growth

Question. Is there an entire function $F$ satisfying first two or all three of the following assertions:
$F(z)\neq 0$ for all $z\in \mathbb{C}$;
$1/F - 1\in H^2(\mathbb{C}_+)$ -- the classical Hardy ...

5
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### Functional equation of bounded analytic functions

Let $\mathbb{D}:=\{z \in \mathbb{C}|~|z|<1\}$. Consider $f,~g \in H^{\infty}(\mathbb{D}):=\{f\in\text{ Hol}(\mathbb{D}): \|f\|_{\infty}:= \sup_{z \in \mathbb{D}} |f(z)| < \infty\}$ such that $f^...

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### Analytic continuation in the Hardy class of analytic functions arcoss the boundary

I have the following question, which I need help addressing.
If $U\subset \mathbb{C}$ and $D\subset\mathbb{C}$, with $\partial U\cap\partial D=\Gamma\neq\emptyset$ of positive measure, are open ...

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### Application of the $\operatorname{BMO}$, $H^1$ duality

Let $f\in \operatorname{BMO}(\partial \Delta)$, then there exists a Carleson measure $\mu$ in $\Delta$ such that
$$f(\zeta)-\int_{\Delta}P_{z}(\zeta)d\mu(z)\in L^{\infty}(\partial \Delta),\ \zeta\in\...

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### Beurling's theorem on invariant subspaces

Beurling's theorem characterize the closed subspaces $M\subset H^2$ of the Hardy space, which are invariant under the shift operator $Sf(z):=zf(z)$, as spaces of the form $\varphi H^2 $ where $\varphi$...