Progressive function
Encyclopedia
In mathematics
, a progressive function ƒ ∈ L2(R) is a function whose Fourier transform
is supported by positive frequencies only:
It is called 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
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a progressive function ƒ ∈ L2(R) is a function whose Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
is supported by positive frequencies only:
It is called regressive if and only if the time reversed function f(−t) is progressive, or equivalently, if
The complex conjugate
Complex conjugate
In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...
of a progressive function is regressive, and vice versa.
The space of progressive functions is sometimes denoted
, which is known as the Hardy spaceHardy space
In complex analysis, the Hardy spaces Hp are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper...
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
.