Progressive function

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

$\mathop {\rm {supp}} {\hat {f}}\subseteq \mathbb {R} _{+}$.

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

$\mathop {\rm {supp}} {\hat {f}}\subseteq \mathbb {R} _{-}$.

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

$f(t)=\int _{0}^{\infty }e^{2\pi ist}{\hat {f}}(s)\ ds$

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

$f(t+iu)=\int _{0}^{\infty }e^{2\pi is(t+iu)}{\hat {f}}(s)\ ds=\int _{0}^{\infty }e^{2\pi ist}e^{-2\pi su}{\hat {f}}(s)\ ds.$

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 $\{t+iu:t,u\in R,u\leq 0\}$.

progressive function at PlanetMath.