Progressive function: Difference between revisions

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In [[mathematics]], a '''progressive function''' ''fƒ''  ∈  ''L''²('''R''') is calleda ''progressive'' [[if and only if]]function itswhose [[Fourier transform]] is supported by positive frequencies only:

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

It is called '''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]] 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>

:<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

:<math>f(t+iu) = \int_0^\infty e^{2\pi i st} e^{-2\pi sus(t+iu)} \hat f(s)\, ds.</math>

:<math>f(t) = \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.