Progressive function
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\}} .
 | This article needs additional citations for verification. (February 2024) |
In mathematics, a progressive function ƒ ∈ L2(R) is a function whose Fourier transform is supported by positive frequencies only:[1]
It is called super regressive if and only if the time reversed function f(−t) is progressive, or equivalently, if
The complex conjugate of a progressive function is regressive, and vice versa.
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
and hence extends to a holomorphic function on the upper half-plane
by the formula
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 .
References
[edit]- ^ Klees, Roland; Haagmans, Roger (6 March 2000). Wavelets in the Geosciences. Springer Science & Business Media. ISBN 978-3-540-66951-7.
This article incorporates material from progressive function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.