Progressive function

In mathematics, a progressive function ƒ ∈ L²(R) is a function whose Fourier transform is supported by positive frequencies only:

It is called super regressive if and only if the time reversed function f(−t) is progressive, or equivalently, if

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

and hence extends to a holomorphic function on the upper half-plane { t + i u : t , u ∈ R , u ≥ 0 }

by the formula

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 { t + i u : t , u ∈ R , u ≤ 0 } .

This article incorporates material from progressive function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.