Progressive function explained

In mathematics, a progressive function ƒ ∈ L²(R) is a function whose Fourier transform is supported by positive frequencies only:^[1]

\rmsupp \hat{f} \subseteq R _{+.}

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

\rmsupp \hat{f} \subseteq 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 formulaf(t) = \int ^{inf ty} _{0} e ^{2\pi} \hat f(s) ds \{ t + iu: t, u \in R, u \geq 0 \}

by the formula

f(t + iu) = \int ^{inf ty} _{0} e ^{2\pi} \hat f(s) ds = \int ^{inf ty} _{0} e ^{2\pi} e ^{-2\pi} \hat f(s) ds.

Conversely, every holomorphic function on the upper half-plane which is uniformly square-integrable on every horizontal linewill 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 \} .

Notes and References

1. Book: Klees . Roland . Haagmans . Roger . Wavelets in the Geosciences . 6 March 2000 . Springer Science & Business Media . 978-3-540-66951-7 . en.