Hardy space

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In complex analysis, the Hardy spaces are analogues of the Lp spaces of functional analysis. They are named for G. H. Hardy.

For example, for spaces of holomorphic functions on the open unit disc, the Hardy space H² consists of the functions f whose mean square value on the circle of radius r remains finite as r → 1 from below.

More generally, the Hardy space H^p for $0<p<\infty$ is the class of holomorphic functions on the open unit disc satisfying

$\sup_{0<r<1} \left(\frac{1}{2\pi} \int_0^{2\pi} \left[f(re^{i\theta})\right]^p \; d\theta\right)^\frac{1}{p}<\infty.$

The number on the left side of the above inequality is the Hardy space $p$ -norm for $f$ , denoted by $\|f\|_{H^p}$ .

For $0<p<q<\infty$ , it can be shown that H^q is a subset of H^p.

1 Applications
2 Factorization
3 Other domains
4 See also
5 References

[edit] Applications

Such spaces have a number of applications in mathematical analysis itself, and also to control theory and scattering theory. A space H² may sit naturally inside an L² space as a 'causal' part, for example represented by infinite sequences indexed by N, where L² consists of bi-infinite sequences indexed by Z.

[edit] Factorization

For $p\geq 1$ , every function $f \in H^p$ can be written as the product f = Gh where G is an outer function and h is an inner function, as defined below.

One says that h(z) is an inner (interior) function if and only if $|h(z)|\leq 1$ on the unit disc and the limit

$\lim_{r\rightarrow 1^-} h(re^{i\theta})$

exists for almost all $θ$ and its modulus is equal to 1.

One says that G(z) is an outer (exterior) function if it takes the form

$G(z)=\exp\left[i\phi+\frac{1}{2\pi} \int_0^{2\pi} \frac{e^{i\theta}+z}{e^{i\theta}-z} g(e^{i\theta}) d\theta \right]$

for some real value $\,\phi$ and some real-valued function g(z) that is integrable on the unit circle.

The inner function can be further factored into a form involving a Blaschke product.

[edit] Other domains

It is possible to define Hardy spaces on other domains than the disc, and in many applications Hardy spaces on a complex half-plane (usually the right half-plane or upper half-plane) are used. See, for example, the cited book of Hoffman.

[edit] See also

H infinity

[edit] References

Joseph A. Cima and William T. Ross, The Backward Shift on the Hardy Space, (2000) American Mathematical Society. ISBN 0-8218-2083-4
Peter Colwell, Blaschke Products - Bounded Analytic Functions (1985), University of Michigan Press, Ann Arbor. ISBN 0-472-10065-3
P. Duren, Theory of $H p$ -Spaces (1970), Academic Press, New York.
Kenneth Hoffman, Banach spaces of analytic functions, (1988), Dover Publications, New York. ISBN 0-486-65785-X