# 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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### 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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### 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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### 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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### 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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### 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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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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### 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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### 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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### 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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### 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 $H^p$ ...
### 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 ...