Progressive function

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

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

It is called super 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 ist}{\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 is(t+iu)}{\hat {f}}(s)\,ds=\int _{0}^{\infty }e^{2\pi ist}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 $\{t+iu:t,u\in R,u\leq 0\}$ .

References

^ Klees, Roland; Haagmans, Roger (6 March 2000). Wavelets in the Geosciences. Springer Science & Business Media. ISBN 978-3-540-66951-7.

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