Progressive function

In mathematics, a function f ∈ L2(R) is called progressive iff its Fourier transform is supported by positive frequencies only:

Related Topics:
Mathematics - Iff - Fourier transform

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:mathop{ m supp}hat{f} subseteq mathbb{R}_+.

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It is called regressive iff the time reversed function f(−t) is progressive, or equivalently, if

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:mathop{ m supp}hat{f} subseteq mathbb{R}_-.

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The complex conjugate of a progressive function is regressive, and vice versa.

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The space of progressive functions is sometimes denoted H^2_+(R), which is known as the Hardy space of the upper half-plane. This is because a progressive function has the Fourier inversion formula

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:f(t) = int_0^infty e^{2pi i st} hat f(s) ds

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and hence extends to a holomorphic function on the upper half-plane { t + iu: t, u in R, u geq 0 }

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by the formula

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:f(t+iu) = int_0^infty e^{2pi i s(t+iu)} hat f(s) ds

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