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:

    “There is, I think, no point in the philosophy of progressive education which is sounder than its emphasis upon the importance of the participation of the learner in the formation of the purposes which direct his activities in the learning process, just as there is no defect in traditional education greater than its failure to secure the active cooperation of the pupil in construction of the purposes involved in his studying.”
    —John Dewey (1859–1952)

    “The information links are like nerves that pervade and help to animate the human organism. The sensors and monitors are analogous to the human senses that put us in touch with the world. Data bases correspond to memory; the information processors perform the function of human reasoning and comprehension. Once the postmodern infrastructure is reasonably integrated, it will greatly exceed human intelligence in reach, acuity, capacity, and precision.”
    —Albert Borgman, U.S. educator, author. Crossing the Postmodern Divide, ch. 4, University of Chicago Press (1992)