Progressive function

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In mathematics, a function f ∈ L2(R) is called progressive if and only if its Fourier transform is supported by positive frequencies only:

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

It is called regressive if and only if the time reversed function f(−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 \}.

This article incorporates material from progressive function on PlanetMath, which is licensed under the GFDL.