Positive harmonic function: Difference between revisions

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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 definite function on a group|positive definiteness]] of their [[Taylor coefficient]]s.

==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

:<math> f(re^{i\theta})=\int_0^{2\pi} {1-r^2\over 1-2r\cos (\theta-\varphi) + r^2} \, d\mu(\varphi).</math>

The formula clearly defines a positive harmonic function with ''f''(0)&nbsp;=&nbsp;1.

Conversely if ''f'' is positive and harmonic and ''r''_''n'' increases to&nbsp;1, define

:<math> f_n(z)=f(r_nz). \, </math>

Then

:<math> f_n(re^{i\theta}) = {1\over 2\pi}\int_0^{2\pi} {1-r^2\over 1-2r\cos(\theta-\varphi) + r^2 }\, f_n(\varphi)\,d\varphi =\int_0^{2\pi}{1-r^2\over 1-2r\cos(\theta-\varphi) + r^2 } d\mu_n(\varphi)</math>

where

:<math> d\mu_n(\varphi)={1\over 2\pi} f(r_n e^{i\varphi})\,d\varphi</math>

is a probability measure.

By a compactness argument (or equivalently in this case

[[Helly's selection theorem]] for [[Stieltjes integral]]s), 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

:<math> f(z) =\int_0^{2\pi} {1 + e^{-i\theta}z\over 1 -e^{-i\theta}z} \, d\mu(\theta).</math>

This follows from the previous theorem because:

* the Poisson kernel is the real part of the integrand above

* the real part of a holomorphic function is harmonic and determines the holomorphic function up to addition of a scalar

* the above formula defines a holomorphic function, the real part of which is given by the previous theorem

==Carathéodory's positivity criterion for holomorphic functions==

Let

:<math> f(z)=1 + b_1 z + b_2 z^2 + \cdots</math>

be a holomorphic function on the unit disk. Then ''f''(''z'') has positive real part on the disk

if and only if

:<math> \sum_m\sum_n a_{m-n} \lambda_m\overline{\lambda_n} \ge 0</math>

for any complex numbers λ₀, λ₁, ..., λ_''N'', where

:<math> a_0=2,\,\,\, a_{-m} =\overline{a_m}</math>

for ''m'' > 0.

In fact from the Herglotz representation for ''n'' > 0

:<math> a_n =2\int_0^{2\pi} e^{-in\theta}\, d\mu(\theta).</math>

Hence

:<math>\sum_m\sum_n a_{m-n} \lambda_m\overline{\lambda_n} =\int_0^{2\pi} \left|\sum_{n} \lambda_n e^{-in\theta}\right|^2 \, d\mu(\theta) \ge 0.</math>

Conversely, setting λ_''n''&nbsp;=&nbsp;''z''^''n'',

:<math>\sum_{m=0}^\infty\sum_{n=0}^\infty a_{m-n} \lambda_m\overline{\lambda_n} = 2(1-|z|^2) \,\Re\, f(z).</math>

==See also==

*[[Bochner's theorem]]

==References==

*{{citation|title=Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen|journal= Math. Ann.|year=1907|volume= 64|pages=95–115|first=C.|last=Carathéodory|doi=10.1007/bf01449883|s2cid= 116695038|url=https://zenodo.org/record/1428260}}

*{{citation|last=Duren|first=P. L.|Duren, Peter L.|title=

*{{citation|last=Duren|first=P. L.|title=Univalent functions|series=Grundlehren der Mathematischen Wissenschaften|volume= 259|publisher= Springer-Verlag|year= 1983|isbn= 0-387-90795-5}}

*{{citation|last=Herglotz|first=G.|title=Über Potenzreihen mit positivem, reellen Teil im Einheitskreis|journal=Ber. Verh. Sachs. Akad. Wiss. Leipzig|volume=63|pages= 501–511|year=1911}}

*{{citation|last=Pommerenke|first= C.|authorlink=Christian Pommerenke|title=Univalent functions, with a chapter on quadratic differentials by Gerd Jensen|series= Studia Mathematica/Mathematische Lehrbücher|volume=15|publisher= Vandenhoeck & Ruprecht|year= 1975}}

*{{citation|last=Riesz|first=F.|title=Sur certains systèmes singuliers d'équations intégrale|journal=Ann. Sci. Éc. Norm. Supér.|volume=28|pages= 33–62|year=1911|doi=10.24033/asens.633|doi-access=free}}

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[[Category:Harmonic analysis]]

[[Category:Complex analysis]]

[[Category:Harmonic functions]]