title: Progressive function
text: # Progressive function
  
  In mathematics, a progressive function ƒ ∈ $L^2(R)$ is a function whose Fourier transform is supported by positive frequencies only:
  
      $\mathop {\rm {supp}} {\hat {f}}\subseteq \mathbb {R} _{+}.$
  
  It is called super regressive if and only if the time reversed function f(−t) is progressive, or equivalently, if
  
      $\mathop {\rm {supp}} {\hat {f}}\subseteq \mathbb {R} _{-}.$
  
  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 $\{t+iu:t,u\in R,u\leq 0\}$.
  
  This article incorporates material from progressive function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
  
  