Progressive Function

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

It is called 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

$f(t+iu) = \int_0^\infty e^{2\pi i s(t+iu)} \hat f(s)\, ds = \int_0^\infty e^{2\pi i st} e^{-2\pi su} \hat f(s)\, ds.$

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 .

This article incorporates material from progressive function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

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