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
\( f(re^{{i\theta }})=\int _{0}^{{2\pi }}{1-r^{2} \over 1-2r\cos(\theta -\varphi )+r^{2}}\,d\mu (\varphi ). \)
The formula clearly defines a positive harmonic function with f(0) = 1.
Conversely if f is positive and harmonic and rn increases to 1, define
\( f_{n}(z)=f(r_{n}z).\, \)
Then
\( f_{n}(re^{{i\theta }})={1 \over 2\pi }\int _{0}^{{2\pi }}{1-r^{2} \over 1-2r\cos(\theta -\varphi )+r^{2}}\,f_{n}(\varphi )\,d\phi =\int _{0}^{{2\pi }}{1-r^{2} \over 1-2r\cos(\theta -\varphi )+r^{2}}d\mu _{n}(\varphi ) \)
where
\( d\mu _{n}(\varphi )={1 \over 2\pi }f(r_{n}e^{{i\varphi }})\,d\varphi \)
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 rn increases to 1, so that fn(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
\( f(z)=\int _{0}^{{2\pi }}{1+e^{{-i\theta }}z \over 1-e^{{-i\theta }}z}\,d\mu (\theta ). \)
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
\( f(z)=1+a_{1}z+a_{2}z^{2}+\cdots \)
be a holomorphic function on the unit disk. Then f(z) has positive real part on the disk if and only if
\( \sum _{m}\sum _{n}a_{{m-n}}\lambda _{m}\overline {\lambda _{n}}\geq 0 \)
for any complex numbers λ0, λ1, ..., λN, where
\( a_{0}=2,\,\,\,a_{{-m}}=\overline {a_{m}} \)
for m > 0.
In fact from the Herglotz representation for n > 0
\( a_{n}=2\int _{0}^{{2\pi }}e^{{-in\theta }}\,d\mu (\theta ).
Hence
\( \sum _{m}\sum _{n}a_{{m-n}}\lambda _{m}\overline {\lambda _{n}}=\int _{0}^{{2\pi }}\left|\sum _{{n}}\lambda _{n}e^{{-in\theta }}\right|^{2}\,d\mu (\theta )\geq 0. \)
Conversely, setting λn = zn,
\( \sum _{{m=0}}^{\infty }\sum _{{n=0}}^{\infty }a_{{m-n}}\lambda _{m}\overline {\lambda _{n}}=2(1-|z|^{2})\,\Re \,f(z). \)
See also
Bochner's theorem
References
Carathéodory, C. (1907), "Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen" (PDF), Math. Ann., 64: 95–115, doi:10.1007/bf01449883
Duren, P. L. (1983), Univalent functions, Grundlehren der Mathematischen Wissenschaften, 259, Springer-Verlag, ISBN 0-387-90795-5
Herglotz, G. (1911), "Über Potenzreihen mit positivem, reellen Teil im Einheitskreis", Ber. Verh. Sachs. Akad. Wiss. Leipzig, 63: 501–511
Pommerenke, C. (1975), Univalent functions, with a chapter on quadratic differentials by Gerd Jensen, Studia Mathematica/Mathematische Lehrbücher, 15, Vandenhoeck & Ruprecht
Riesz, F. (1911), "Sur certains systèmes singuliers d'équations intégrale", Ann. Sci. Éc. Norm. Supér., 28: 33–62, doi:10.24033/asens.633
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
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