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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 representation theorem'', was proved by [[Gustav Herglotz]] 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 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
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:<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) = 1.
Conversely if ''f'' is positive and harmonic and ''r''<sub>''n''</sub> increases to 1, define
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:<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''<sub>''n''</sub> increases to 1, so that ''f''<sub>''n''</sub>(''z'') tends to ''f''(''z''), the Herglotz formula follows.
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==Carathéodory's positivity criterion for holomorphic functions==
Let
:<math> f(z)=1 + a_1 z + a_2 z^2 + \cdots</math>
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:<math> \sum_m\sum_n a_{m-n} \lambda_m\overline{\lambda_n} \ge 0</math>
for any complex numbers λ<sub>0</sub>, λ<sub>1</sub>, ..., λ<sub>''N''</sub>, where
:<math> a_0=2,\,\,\, a_{-m} =\overline{a_m}</math>
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:<math>\sum_m\sum_n a_{m-n} \lambda_m\overline{\lambda_n} =\int_0^{2\pi} \left|\sum_{n=}^N \lambda_n e^{-in\theta}\right|^2 \, d\mu(\theta) \ge 0.</math>
Conversely, setting λ<sub>''n''</sub> = ''z''<sup>''n''</sup>,
:<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>
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==References==
*{{citation|title=Über den Variabilitatsbereich der Koeffizienten von Potenzreflien, die gegebene Werte nicht annehmen|journal= Math. Ann.|year=1907|volume= 64|pages=95–115|first=C.|last=Carathéodory}}
*{{citation|last=Duren|first=P.
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}}
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