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.

Famous quotes containing the words progressive and/or function:

    “The self ... might be regarded as a sort of citadel of the mind, fortified without and containing selected treasures within, while love is an undivided share in the rest of the universe. In a healthy mind each contributes to the growth of the other: what we love intensely or for a long time we are likely to bring within the citadel, and to assert as part of ourself. On the other hand, it is only on the basis of a substantial self that a person is capable of progressive sympathy or love.”
    —Charles Horton Cooley (1864–1929)

    “As a medium of exchange,... worrying regulates intimacy, and it is often an appropriate response to ordinary demands that begin to feel excessive. But from a modernized Freudian view, worrying—as a reflex response to demand—never puts the self or the objects of its interest into question, and that is precisely its function in psychic life. It domesticates self-doubt.”
    —Adam Phillips, British child psychoanalyst. “Worrying and Its Discontents,” in On Kissing, Tickling, and Being Bored, p. 58, Harvard University Press (1993)