In mathematics, a progressive function ƒ ∈ L<sup>2</sup>(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 , 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.
Regressive functions are similarly associated with the Hardy space on the lower half-plane .