mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a progressive function ''ƒ'' ∈ ''L''²(R) is a function whose

Fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...

is supported by positive frequencies only: :

\mathop\hat \subseteq \mathbb_+.

It is called super regressive if and only if the time reversed function ''f''(−''t'') is progressive, or equivalently, if :

\mathop\hat \subseteq \mathbb_-.

The

complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - ...

of a progressive function is regressive, and vice versa. The space of progressive functions is sometimes denoted

H^2_+(R)

, which is known as the

Hardy space In complex analysis, the Hardy spaces (or Hardy classes) H^p are spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper . In real anal ...

of the upper half-plane. This is because a progressive function has the Fourier inversion formula :

f(t) = \int_0^\infty e^ \hat f(s)\, ds

and hence extends to a

holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...

on the

upper half-plane In mathematics, the upper half-plane, is the set of points in the Cartesian plane with The lower half-plane is the set of points with instead. Arbitrary oriented half-planes can be obtained via a planar rotation. Half-planes are an example ...

\

by the formula :

f(t+iu) = \int_0^\infty e^ \hat f(s)\, ds
= \int_0^\infty e^ e^ \hat f(s)\, ds.

Conversely, every holomorphic function on the upper half-plane which is uniformly square-integrable on every horizontal line will arise in this manner. Regressive functions are similarly associated with the Hardy space on the lower half-plane

\

References

{{PlanetMath attribution, id=5993, title=progressive function Hardy spaces Types of functions