Progressive function

In mathematics, a function "f" ∈ "L"²(R) is called "progressive" if and only if its Fourier transform is supported by positive frequencies only:

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

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

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

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

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:$f(t)\; =\; int\_0^infty\; e^\{2pi\; i\; st\}\; hat\; f(s)\; ds$and hence extends to a holomorphic function on the upper half-plane $\{\; t\; +\; iu:\; t,\; u\; in\; R,\; u\; geq\; 0\; \}$by the formula:$f(t+iu)\; =\; int\_0^infty\; e^\{2pi\; i\; s(t+iu)\}\; hat\; f(s)\; ds=\; int\_0^infty\; e^\{2pi\; i\; st\}\; e^\{-2pi\; su\}\; hat\; f(s)\; ds.$Conversely, every holomorphic function on the upper half-plane which is uniformly square-integrable on every horizontal linewill arise in this manner.

Regressive functions are similarly associated with the Hardy space on the lower half-plane $\{\; t\; +\; iu:\; t,\; u\; in\; R,\; u\; leq\; 0\; \}$.