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 Hp for 
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.](a/../upload/math/0fcbd5213fa8807e804f554fdb5d98e8.png)
The number on the left side of the above equation is the Hardy space p-norm for f, denoted by 
.
For 
, it can be shown that Hq is a subset of Hp.
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.
Factorization
For 
, every function 
can be written as the product f = G h where G is an outer function and h is an inner function, defined below.
One says that h(z) is an inner function if and only if
on the unit disc and the limit
exists for almost all θ.
One says that G(z) is an outer 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]](a/../upload/math/96ad04b29df56f06e2de779f081c37f2.png)
for some real value φ and some real-valued function g(z) that is integrable on the unit circle.
The inner function can be further factored into form involving a Blaschke product.
See also
References
- Joeseph A. Cima and William T. Ross, The Backwards 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