Chemistry - Hardy space

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 equation 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.

Contents

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 $p\geq 1$ , every function $f \in H^p$ 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 $|h(z)|\leq 1$ on the unit disc and the limit $\lim_{r\rightarrow 1^-} h(re^{i\theta})$ exists for almost all $θ$ .

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

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

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.

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

Categories: Complex analysis