Progressive function

In mathematics, a function "f" ∈ "L"²(R) is called "progressive" if and only if its Fourier transform is supported by positive frequencies only:

:$$mathop{ m supp}hat{f} subseteq mathbb{R}_+.

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

:$$mathop{ m supp}hat{f} subseteq mathbb{R}_-.

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^{2pi i st} hat f(s) dsand hence extends to a holomorphic function on the upper half-plane $${ t + iu: t, u in R, u geq 0 }by the formula:$$f(t+iu) = int_0^infty e^{2pi i s(t+iu)} hat f(s) ds= int_0^infty e^{2pi i st} e^{-2pi su} hat f(s) ds.Conversely, every holomorphic function on the upper half-plane which is uniformly square-integrable on every horizontal linewill arise in this manner.

Regressive functions are similarly associated with the Hardy space on the lower half-plane $${ t + iu: t, u in R, u leq 0 }.