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

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How to prove $f$ is outer, when $Re f$ >0?

This is an exercise problem, which can be found in most of the book which deals with Hardy Space. Any kind of suggestion or hint is also welcome. The question is following: Let $f$ be a function in ...
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When the singular inner part disappear in inner outer factorization?

I saw this remark in Hoffman's book - "Banach space of analytic function". If $f$ is analytic in a neighborhood of $\bar{\mathbb{D}}$, the closure of $\mathbb{D}$; then in the inner-outer ...
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How to check if a matrix transfer function is in Hardy-infinity space?

Just like the question says. For instance if I have a matrix transfer function $$\mathbf{G}(s) = \begin{bmatrix}s & -s \\ T & s \\ \end{bmatrix}$$ where $T$ is a positive constant, how can I ...
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Find a sequence of complex polynomials with certain properties. (Hardy spaces over unit circle)

Let $\lambda\in \Bbb S^1$. Find a sequence of complex polynomials $p_n(z)$ such that for any $c>0$ the following inequality does not hold: $$|p_n(\lambda)|\le c\cdot \|p_n\|$$ where ...
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Show that for every real-valued $L^2$ function $u$ on $S^1$ there is a real-valued $v$ in the same space such that $u + iv\in \widetilde{\mathbf H}^2$

For a homework exercise ($1.8$ in the book An Introduction to Operators on the Hardy-Hilbert Space) I am asked to show Let $u$ be a real-valued function in $L^2(S^1)$. Show that there exists a ...
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How to decide whether a function is in $H^p_+$.

$H^p_+$ is the Hardy space on the upper half-plane of $\mathbb C$ consisting of functions $f$ satisfying $$ \sup_{y_0>0}\left(\int_{-\infty}^{\infty} |f(x+iy_0)|^p dx\right)^{\frac 1p}<\infty $$ ...
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Interpolation sequences and open mapping theorem

I'm using Garnett's "Bounded Analytic Functions" as a course text and looking at interpolation sequences. $z_n$ is a sequence of interpolation if for each sequence $a_n \in l^{\infty}$ there exists $f ...
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A function in $H^\infty(\Delta^2)$

Can you give an example of a function $f\in H^\infty(\Delta^2)$ with $f^{-1}\in L^\infty(T^2)$ but not inner? Here $H^\infty(\Delta^2)$ is the space of all bounded analytic functions defined on ...
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Bounded-below multiplication operator on Hardy space

Let $H^2(\Delta^2)$ denotes the Hardy space on the bi-disc $\Delta^2$ and $M_f :H^2(\Delta^2)\rightarrow H^2(\Delta^2)$ be multiplication operator by $f\in H^\infty(\Delta^2)$ defined by ...
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Multiplication operator is bijective?

Let $H^2(\Delta^2)$ denote the Hardy space on the polydics and $M_f:H^2(\Delta^2)\rightarrow H^2(\Delta^2)$ be a multiplication operator by $f\in H^\infty(\Delta^2)$. Is the operator is bijective? ...
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A simple function with no tangential limits but with non-tangential limits

I am going to be teaching a course about the Hardy space and I would like to show the students that the non-tangential limit is a necessary concept BEFORE telling them about Blaschke products. Is ...
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Literature on interpolation in Hardy spaces

I'm an undergraduate mathematics student and I'm searching for notes and books on Hardy spaces $H^p$, in particular interpolation theory including topics like Carleson measures, Carleson's $H^\infty$ ...
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Showing a certain sequence is an orthonormal basis of $H^2(\mathbb{R}_{+}^{2}).$

The problem is to show $$\left\{\frac{1}{\pi^{1/2}(i+z)}\left(\frac{i-z}{i+z}\right)^n\right\}_{n=1}^{\infty}$$ is an orthonormal basis of $H^2(\mathbb{R}_{+}^{2}).$ In another exercise, I have ...
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The definition of Hardy space

The definition of the Hardy space consists functions $u(z)$ that are analytic a. outside the closed unit disc or b. inside the open unit disc What is the difference between the two definition ...
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Real Hardy space question

Please see this related question for the definition of the grand maximal function and the class of normalised test functions $\mathcal{T}$. I will refer to them in this question. Let $f\in ...
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Constructing a specific normalised test function.

Given the following class of normalised test functions: \begin{equation} \mathcal{T}=\{\phi\in C_c^{\infty}(\mathbb{R}^n)\ |\ \text{supp }\phi \subseteq B(0, 1),\ \|D\phi\|_{\infty}\leq 1\} ...
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Hardy Space Cancellation Condition

I have been reading the chapter on Hardy spaces in Stein's Harmonic Analysis book, and I am having a lot of trouble figuring something out. The setting here is $\mathbb{R}^n.$ Let $f \in L^q$ be ...
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$L^1(\mathbb{R}^n)$ functions not in $\mathcal{H}^1(\mathbb{R}^n)$

I am wondering how to imagine the Hardy space on $\mathcal{H}^1(\mathbb{R}^n)$ and in particular what sort of functions are in $L^1(\mathbb{R}^n)\backslash\mathcal{H}^1(\mathbb{R}^n)$. Furthermore, is ...
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Why p>1, $L^p$ and $H^p$ are essentially the same?

the conclusion comes from http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183538894 (the first page) $L^{p}$ is Lebesgue integral function space, ...
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Showing atomic $H^{1,p}$ is a Banach Space

Consider $1<p\leq\infty$ and let $H^{1,p}=H^{1,p}(\mathbb{R}^n)$ be an atomic Hardy space, that is $H^{1,p}\subset L^1(\mathbb{R}^n)$, and $f\in H^{1,p}$ if there exists a sequence of $p$-atoms ...
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Composition of $\mathrm H^p$ function with Möbius transform

Let $f:\mathbb D\rightarrow \mathbb C$ be a function in $\mathrm{H}^p$, i.e. $$\exists M>0,\text{ such that }\int_0^{2\pi}|f(re^{it})|^pdt\leq M<\infty,\forall r\in[o,1)$$ Consider a Möbius ...
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Prove that $S$is a closed subspace of $H^2$ invariant under multiplication by $z$. Find the inner function $F$ such that $S=FH^2$

Let ${\alpha_n}$ be a sequence of points in the open unit disc such that $\sum(1-|\alpha_n|)<\infty$. Let $S$ be the set of all functions $f$ in $H^2$ spaces such that $f(\alpha_n)=f'(\alpha_n)=0$ ...
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How to show that $ν(z)$ is Carleson measure

If $ν(z)=|1+z|^ β dμ(z)$, $β\in \!\ R^-$. How to show that $ν(z)$ is Carleson? I know that $ν$ is Carleson measure if $ν(Q_I)\leqq \!\ c.I$ But how to apply this?
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Hardy space question

Let $T$ be the unit circle. Let $\phi\in C(T)$ and let $\psi$ be a function in $L^2(T)$ such that $\phi+i\psi\in H^2$. Assume both $\psi$ and $\phi$ are real-valued. Show $e^{\phi+i\psi}\in ...
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invariant subspace of a Hardy space

Let $T$ be the unit circle and $H^1=\{f\in L^1(T): \int_0^{2\pi} f(e^{it})\chi_n(e^{it})dt=0 \text{ for } n>0\}$ where $\chi_n(e^{it})=e^{int}$. Let $M$ be a closed subspace of $H^1$. Then ...
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Removable sets for harmonic functions and Hardy spaces of general domains

Let $\Omega$ be a domain of the complex plane. The Hardy space $H^p(\Omega)$ is defined, for $1 \leq p<\infty$, as the class of functions $f$ that are holomorphic on $\Omega$ such that $|f|^p$ has ...