In mathematics, a positive harmonic function on the unit disc in the complex numbers is characterized as the Poisson integral of a finite positive measure on the circle. This result, the "Herglotz-Riesz representation theorem", was proved independently by Gustav Herglotz and Frigyes Riesz in 1911. It can be used to give a related formula and characterization for any holomorphic function on the unit disc with positive real part. Such functions had already been characterized in 1907 by Constantin Carathéodory in terms of the positive definiteness of their Taylor coefficients.
Herglotz-Riesz representation theorem for harmonic functions.
A positive function "f" on the unit disk with "f"(0) = 1 is harmonic if and only if there is a probability measure μ on the unit circle such that
formula_1
The formula clearly defines a positive harmonic function with "f"(0) = 1.
Conversely if "f" is positive and harmonic and "r""n" increases to 1, define
formula_2
Then
formula_3
where
formula_4
is a probability measure.
By a compactness argument (or equivalently in this case
Helly's selection theorem for Stieltjes integrals), a subsequence of these probability measures has a weak limit which is also a probability measure μ.
Since "r""n" increases to 1, so that "f""n"("z") tends to "f"("z"), the Herglotz formula follows.
Herglotz-Riesz representation theorem for holomorphic functions.
A holomorphic function "f" on the unit disk with "f"(0) = 1 has positive real part if and only if there is a probability measure μ on the unit circle such that
formula_5
This follows from the previous theorem because:
Carathéodory's positivity criterion for holomorphic functions.
Let
formula_6
be a holomorphic function on the unit disk. Then "f"("z") has positive real part on the disk
if and only if
formula_7
for any complex numbers λ0, λ1, ..., λ"N", where
formula_8
for "m" ; 0.
In fact from the Herglotz representation for "n" ; 0
formula_9
Hence
formula_10
Conversely, setting λ"n" = "z""n",
formula_11