Progressive function

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

$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\hat{f}\subseteq {\mathbb{ℝ}}_{+}.$

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

$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\hat{f}\subseteq {\mathbb{ℝ}}_{-}.$

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

$f(t)=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{2\pi ist}\hat{f}(s)ds$

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

by the formula

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

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