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In mathematics, a function f ∈ L2(R) is called progressive iff its Fourier transform is supported by positive frequencies only:
- <math>\mathop{\rm supp}\hat{f} \subseteq \mathbb{R}_+<math>.
It is called regressive iff 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>.
This article incorporates material from progressive function (http://planetmath.org/?op=getobj&from=objects&id=5993) on PlanetMath, which is licensed under the GFDL.
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