Progressive function

In mathematics, a progressive function ƒ ∈ L²(R) is a function whose Fourier transform is supported by positive frequencies only:^[1]

${\displaystyle \mathop {\rm {supp}} {\hat {f}}\subseteq \mathbb {R} _{+}.}$

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

${\displaystyle \mathop {\rm {supp}} {\hat {f}}\subseteq \mathbb {R} _{-}.}$

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

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

${\displaystyle f(t)=\int _{0}^{\infty }e^{2\pi ist}{\hat {f}}(s)\,ds}$

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

by the formula

${\displaystyle f(t+iu)=\int _{0}^{\infty }e^{2\pi is(t+iu)}{\hat {f}}(s)\,ds=\int _{0}^{\infty }e^{2\pi ist}e^{-2\pi su}{\hat {f}}(s)\,ds.}$

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 ${\displaystyle \{t+iu:t,u\in R,u\leq 0\}}$.