Hardy space

In the theory of functions a Hardy space is a function space of holomorphic functions on certain subsets of. Hardy spaces are the equivalents of the rooms in the functional analysis. They are named after Godfrey Harold Hardy, who introduced it in 1914.

• 4.1 Definition
• 4.2 Atomic decomposition
• 4.3 Connection to the Hardy spaces
• 4.4 Further properties

Definition

Usually, two classes of Hardy spaces are defined, depending on the area in the complex plane on which their functions are defined.

Hardy spaces on the unit disk

Be the unit disk in. Then there is for the Hardy space of all holomorphic functions for which it holds

The value of the term on the left side of this inequality is referred to as " standard " by, in symbols.

For one set and is defined as the space of all holomorphic functions for which this value is finite.

Hardy spaces on the upper half-plane

Is in the upper half plane. Then there is for the Hardy space of all holomorphic functions for which it holds

The value of the term on the left side of this inequality is also referred to as " standard " by, in symbols.

For one set and is defined as the space of all holomorphic functions for which this value is finite.

If general of Hardy spaces is mentioned, is usually clear what is meant by the two classes ( ie, whether or ); Usually, it is the space of functions on the unit disk.

Factorization

For each function can be written as a product in which an outer and an inner functional function.

For on the unit disk, for example, an inner function if and only if the following applies to the unit disk and the limit

Exists for almost all and its absolute value is equal to 1. is an external function when

For a real value and a real-valued and integrable function on the unit circle.

Other properties

• For the spaces are Banach spaces.
• For true and.

• For true. In this case, all these inclusions are real.

Real Hardy spaces

From the Hardy - spaces of the upper half-plane and Elias stone Guido white developed the theory of real Hardy spaces.

Definition

Be a Schwartz function on and for t > 0 is a Dirac sequence. Is a temperature- distribution, the maximum radial function and the non- maximum tangent function is defined by

Here denotes the convolution between a tempered distribution and a Schwartz function.

Charles Fefferman and Elias M. Stein for proven and that the following three conditions are equivalent:

One defines the real Hardy space as the space containing all tempered distributions that satisfy the above conditions.

Atomic decomposition

In particular, functions have the property that they can be broken down into a series of "small" functions of so-called atoms. An atom is a function so that applies:

The claims 1 and 2 guarantee the inequality and the demand 3 brings the stronger inequality

The theorem on the atomic decomposition says now, for with may as number of atoms

Be written. Here is a sequence of complex numbers. The series converges in the distribution sense, and it is further

Connection to the Hardy - areas

As mentioned above, the real Hardy spaces from the Hardy spaces of function theory out have been developed. This is explained in the following section, however, we limit ourselves here to the case. The interesting case p = 1 is therefore dealt with and you get the whole range.

Be

Functions on the upper half- plane containing the generalized Cauchy- Riemann differential equations

For meet.

Thus, every function is a harmonic function and in the case of the generalized Cauchy- Riemann differential equations correspond exactly to the normal Cauchy -Riemann equations. Thus, there exists a holomorphic function with respect to the variable.

After another set of Fefferman and Stein a harmonic function if and only met one of the three equivalent conditions, if there exists a function which satisfies the generalized Cauchy- Riemann differential equations and which is - limited, which

Means.

Other properties

• For applies analogously. So the real Hardy spaces can be identified with the corresponding spaces for these p.
• In the event you can as a proper subset of understand.
• Is dense in for.
• The Hardy space is not reflexive, the function space BMO is its dual space.

Applications

Hardy spaces are used in the functional analysis itself, but also in the control theory and scattering theory. They play a fundamental role in signal processing. A real-valued signal, that is for all of finite energy, it assigns the analytic signal, so that. If so, and

( The function is the Hilbert transform of ). For example, for a signal, the associated analytic signal is given by.