Progressive function

In mathematics, a progressive function &fnof; &isin; L2(R) is a function whose Fourier transform is supported by positive frequencies only:


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

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


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

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

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


 * $$f(t) = \int_0^\infty e^{2\pi i st} \hat f(s)\, ds$$

and 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^{2\pi i s(t+iu)} \hat f(s)\, ds

= \int_0^\infty e^{2\pi i st} 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 $$\{ t + iu: t, u \in R, u \leq 0 \}$$.