# Progressive function

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In [mathematics](/source/mathematics), a '''progressive function''' ''&fnof;''&nbsp;&isin;&nbsp;''L''<sup>2</sup>('''R''') is a function whose [Fourier transform](/source/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''(&minus;''t'') is progressive, or equivalently, if

:<math>\mathop{\rm supp}\hat{f} \subseteq \mathbb{R}_-.</math>

The [complex conjugate](/source/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](/source/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](/source/holomorphic_function) on the [upper half-plane](/source/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

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Adapted from the Wikipedia article [Progressive function](https://en.wikipedia.org/wiki/Progressive_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Progressive_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.