Progressive function

Template:More citations needed In mathematics, a progressive function ƒ ∈ L²(R) is a function whose Fourier transform is supported by positive frequencies only:<ref>Template:Cite book</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

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