In mathematics, a progressive function 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.

, 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

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.

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