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Encyclopedia > Hardy space

In complex analysis, the Hardy spaces are analogues of the Lp spaces of functional analysis. They are named for G. H. Hardy. Complex analysis is the branch of mathematics investigating holomorphic functions, i. ... In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ... Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ... G. H. Hardy Godfrey Harold Hardy (February 7, 1877 – December 1, 1947) was a prominent British mathematician, known for his achievements in number theory and mathematical analysis. ...

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. Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... A disc of unit radius on a plane is called a unit disc. ... In mathematics, the root mean square or rms is a statistical measure of the magnitude of a varying quantity. ...

More generally, the Hardy space Hp for is the class of holomorphic functions on the open unit disc satisfying

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. A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X...



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. Analysis is the generic name given to any branch of mathematics which depends upon the concepts of limits and convergence, and studies closely related topics such as continuity, integration, differentiability and transcendental functions. ... In engineering and mathematics, control theory deals with the behaviour of dynamical systems over time. ... The scattering theory is a branch of quantum mechanics whose aim is the study of scattering events. ... This is a page about mathematics. ... This is a page about mathematics. ...


For , every function 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 on the unit disc and the limit exists for almost all θ. In mathematics, the phrase almost all has a number of specialised uses. ...

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

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 a form involving a Blaschke product. In mathematics, the Blaschke product in complex analysis is an analytic function designed to have zeros at a (finite or infinite) sequence of prescribed complex numbers a0, a1, ... inside the unit disc. ...

See also

  • H infinity

H infinity or is a method in control theory for the design of optimal controllers. ...


  • 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

  Results from FactBites:
G. H. Hardy at AllExperts (843 words)
Hardy was Sadleirian Professor at Cambridge from 1931 to 1942; he had left Cambridge to take the Savilian Chair of Geometry at Oxford in the aftermath of the Bertrand Russell affair during World War I.
Hardy was more in tune with the cours d'analyse methods dominant in France, and aggressively promoted his conception of pure mathematics, in particular against the hydrodynamics which was an important part of Cambridge mathematics.
Hardy is also known for formulating the Hardy-Weinberg principle, a basic principle of population genetics, independently from Wilhelm Weinberg in 1908.
Wikinfo | Lp space (778 words)
spaces are spaces of p-power integrable functions, and corresponding sequence spaces.
If S is the set of natural numbers, the space is usually denoted as ellp (ie., without the space indication).
The question of whether all Banach spaces have such an embedding was answered negatively by B. Tsirelson's construction of Tsirelson space in 1974.
  More results at FactBites »



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