In mathematics, a progressive function ƒ ∈ L2(R) is a function whose Fourier transform is supported by positive frequencies only:
s u p p f ^ ⊆ R + . {\displaystyle \mathop {\rm {supp}} {\hat {f}}\subseteq \mathbb {R} _{+}.}
It is called super regressive if and only if the time reversed function f(−t) is progressive, or equivalently, if
s u p p f ^ ⊆ R − . {\displaystyle \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 ) {\displaystyle 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 ) = ∫ 0 ∞ e 2 π i s t f ^ ( s ) d s {\displaystyle 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 + i u : t , u ∈ R , u ≥ 0 } {\displaystyle \{t+iu:t,u\in R,u\geq 0\}}
by the formula
f ( t + i u ) = ∫ 0 ∞ e 2 π i s ( t + i u ) f ^ ( s ) d s = ∫ 0 ∞ e 2 π i s t e − 2 π s u f ^ ( s ) d s . {\displaystyle 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 + i u : t , u ∈ R , u ≤ 0 } {\displaystyle \{t+iu:t,u\in R,u\leq 0\}} .
References
This article incorporates material from progressive function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
