Progressive Function

In mathematics, a progressive function ''ƒ'' ∈ ''L''²(R) is a function whose Fourier transform 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 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 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 on the upper half-plane $\$

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 $\$.

{{PlanetMath attribution, id=5993, title=progressive function

Hardy spaces
Types of functions