In mathematics, a function f ∈ L²(R) is called progressive iff its Fourier transform is supported by positive frequencies only: Mathematics is the study of quantity, structure, space and change. ... ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Although P iff Q is most standard, common alternative phrases include Q is necessary and sufficient for P and P... The Fourier transform, named after Jean Baptiste Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. ...

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

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The complex conjugate of a progressive function is regressive, and vice versa. In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ...

The space of progressive functions is sometimes denoted , which is known as the Hardy space of the upper half-plane. This is because a progressive function has the Fourier inversion formula In complex analysis, the Hardy spaces are analogues of the Lp spaces of functional analysis. ...

and hence extends to a holomorphic function on the upper half-plane by the formula Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...

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 .

This article incorporates material from progressive function on PlanetMath, which is licensed under the GFDL. PlanetMath is a free, collaborative, online mathematics encyclopedia. ...