Hardy space

From Academic Kids

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 <math>0
<math>\sup_{0 The number on the left side of the above equation is the Hardy space <math>p<math>-norm for <math>f<math>, denoted by <math>\|f\|_{H^p}<math>. For <math>0q is a subset of Hp.
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 <math>p\geq 1<math>, every function <math>f \in H^p<math> 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 <math>|h(z)|\leq 1<math> on the unit disc and the limit <math>\lim_{r\rightarrow 1^-} h(re^{i\theta})<math> exists for almost all <math>\theta<math>.

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

<math>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]<math> for some real value <math>\phi<math> 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.

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
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