{{More citations needed|date=February 2024}} In mathematics, a '''progressive function''' ''ƒ'' ∈ ''L''<sup>2</sup>('''R''') is a function whose Fourier transform is supported by positive frequencies only:<ref>{{cite book |last1=Klees |first1=Roland |last2=Haagmans |first2=Roger |title=Wavelets in the Geosciences |date=6 March 2000 |publisher=Springer Science & Business Media |isbn=978-3-540-66951-7 |url=https://books.google.com/books?id=6-xW6Bo-UA8C&dq=%22Progressive+function%22+-wikipedia+Fourier+transform&pg=PA13 |language=en}}</ref>
:<math>\mathop{\rm supp}\hat{f} \subseteq \mathbb{R}_+.</math>
It is called '''super regressive''' if and only if the time reversed function ''f''(−''t'') is progressive, or equivalently, if
:<math>\mathop{\rm supp}\hat{f} \subseteq \mathbb{R}_-.</math>
The complex conjugate of a progressive function is regressive, and vice versa.
The space of progressive functions is sometimes denoted <math>H^2_+(R)</math>, which is known as the Hardy space of the upper half-plane. This is because a progressive function has the Fourier inversion formula
:<math>f(t) = \int_0^\infty e^{2\pi i st} \hat f(s)\, ds</math>
and hence extends to a holomorphic function on the upper half-plane <math>\{ t + iu: t, u \in R, u \geq 0 \}</math>
by the formula
:<math>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.</math>
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 <math>\{ t + iu: t, u \in R, u \leq 0 \}</math>.
==References== {{reflist}}
{{PlanetMath attribution|id=5993|title=progressive function}}
Category:Hardy spaces Category:Types of functions