Progressive function

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

[math]\displaystyle{ \mathop{\rm supp}\hat{f} \subseteq \mathbb{R}_+. }[/math]

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

[math]\displaystyle{ \mathop{\rm supp}\hat{f} \subseteq \mathbb{R}_-. }[/math]

The complex conjugate of a progressive function is regressive, and vice versa.

The space of progressive functions is sometimes denoted [math]\displaystyle{ H^2_+(R) }[/math], which is known as the Hardy space of the upper half-plane. This is because a progressive function has the Fourier inversion formula

[math]\displaystyle{ f(t) = \int_0^\infty e^{2\pi i st} \hat f(s)\, ds }[/math]

and hence extends to a holomorphic function on the upper half-plane [math]\displaystyle{ \{ t + iu: t, u \in R, u \geq 0 \} }[/math]

by the formula

[math]\displaystyle{ f(t+iu) = \int_0^\infty e^{2\pi i s(t+iu)} \hat f(s)\, ds = \int_0^\infty e^{2\pi i st} e^{-2\pi su} \hat f(s)\, ds. }[/math]

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 [math]\displaystyle{ \{ t + iu: t, u \in R, u \leq 0 \} }[/math].