Hardy Space

Hardy space is a special kind of functional spaces in complex analysis , an analog $L^{p}$ ${\ displaystyle L ^ {p}}$ $L ^ {p}$ -spaces from functional analysis . Named after the English mathematician Hardy .

Definition

Hardy Space $\ H^{p}$ ${\ displaystyle \ H ^ {p}}$ at ${\displaystyle 0<semantics><mrow class=$ ${\ displaystyle 0 <p <\ infty}$ Is a class of holomorphic functions on an open unit disk on the complex plane , satisfying the following condition

\sup _{0<r<1}\left({\frac {1}{2\pi }}\int \limits _{0}^{2\pi }\left|f(re^{i\theta })\right|^{p}\;d\theta \right)^{\frac {1}{p}}<\infty .

{\ displaystyle \ sup _ {0 <r <1} \ left ({\ frac {1} {2 \ pi}} \ int \ limits _ {0} ^ {2 \ pi} \ left | f (re ^ { i \ theta}) \ right | ^ {p} \; d \ theta \ right) ^ {\ frac {1} {p}} <\ infty.}

The left side of this inequality is called $\ p$ ${\ displaystyle \ p}$ - the norm in Hardy space or just the Hardy norm for $\ f$ ${\ displaystyle \ f}$ , and is denoted $\ |f|_{H^{p}}$ ${\ displaystyle \ | f | _ {H ^ {p}}}$ . As is the case $L^{p}$ ${\ displaystyle L ^ {p}}$ -spaces, this norm is generalized to the case $p=\infty$ ${\ displaystyle p = \ infty}$ as

|f|_{H^{\infty }}\ =\ \sup _{0<r<1}\sup _{z:\ |z|=r}|f(z)|\ =\ \sup _{z:\ |z|<1}|f(z)|.

{\ displaystyle | f | _ {H ^ {\ infty}} \ = \ \ sup _ {0 <r <1} \ sup _ {z: \ | z | = r} | f (z) | \ = \ \ sup _ {z: \ | z | <1} | f (z) |.}

For case ${\displaystyle 0<semantics><mrow class=$ ${\ displaystyle 0 <p <q \ leq \ infty}$ can show that $\ H^{q}$ ${\ displaystyle \ H ^ {q}}$ is a subset of the set $\ H^{p}$ ${\ displaystyle \ H ^ {p}}$ .

Applications

Such spaces are used both in classical mathematical analysis and in other branches of analysis and its applications, for example, harmonic analysis , control theory (in particular, for the synthesis of robust control systems ) and dispersion theory .

Hardy Space

Definition

Applications

See also

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