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 H^p 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 H^q is a subset of H^p. 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...

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. 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. ...

Factorization

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. ...

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