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 Hp for 0 is the class of holomorphic functions on the open unit disc satisfying
\sup_{0f(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, 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 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 \theta. One says that G(z) is an outer function if it takes the form
G(z)=\exp\left\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 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

 

<< PreviousWord BrowserNext >>
ramat ha sharon
jadis
environmental health
longreach, queensland
erectile tissue
earle brown
autosuggestion
william lane craig
roman province
christ.
benbecula records
list of ancient romans
malcolm turnbull
hume's principle
graupel
lake geneva (disambiguation)
amami woodcock
dusky woodcock
taran
sulawesi woodcock
moluccan woodcock
villers cotterts
mary gaudron
villers bocage, calvados
mr. baseball
hula hoop
st mary's
edward adolphus ferdinand seymour, earl st. maur
clostridium
patrick logan
operation defensive shield
family matters
in vitro toxicology
university of zurich
white rumped sandpiper
george chapman
geographic coding
toynbee tiles
john feckenham
baird's sandpiper
john hooper
khoikhoi
clonakilty
nicholas heath
Copyright 2005-2009 OnPedia.com. All Rights Reserved