# 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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### $L^p$ norm of $\frac{1}{1-\overline{\alpha }e^{i\theta}}.$

I am trying to calculate the $L^p$ norm of the function $f_\alpha(e^{i\theta}) = \frac{1}{1-\overline{\alpha }e^{i\theta}},$ where $\alpha$ is a complex number with $|\alpha| < 1.$ Id est, for ...
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### Proof that polynomials are dense in $H^p$ spaces

In some papers and books I read they used that the polynomials are dense in $H^p$ for $1\le p<\infty$, but I could not find a proof for this stament. I tried to proof it myself but failed. My idea ...
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### Hardy space and boundary regularity

Let $H^\infty(D)$ be the Hardy-space of bounded, holomorphic functions $\Phi:D\rightarrow \mathbb{C}$, defined on the unit disk $D\subset \mathbb{C}$. It is a standard result that $\Phi\in H^\infty(D)$...
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### Is $\sqrt{i \cdot \frac{1+z}{1-z}}$ in the Hardy space $H^1(\mathbb{D})$

I am trying to prove that $$\sqrt{i \cdot \frac{1+z}{1-z}}$$ is in the Hardy space $H^1(\mathbb{D})$, where we take the principal branch of the square root. I think that this is true but I am stuck at ...
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### Prove a norm equality (Composition operator on Hardy space)

For $\lambda\in \mathbb{D}$ fixed, let $C_{\phi}$ denote the composition operator on Hardy space $H^2(\mathbb{D})$ (I.e. $C_{\phi}f:=f(\phi)$) with symbol $\phi=\frac{\lambda-z}{1-\bar{\lambda}z}$ (a ...
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### 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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### 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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### 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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### 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 ...
### 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 ...