Herglotz-Riesz representation theorem for harmonic functionsedit
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
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
Then
where
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 functionsedit
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
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 functionsedit
Let
be a holomorphic function on the unit disk. Then f(z) has positive real part on the disk if and only if
for any complex numbers λ0, λ1, ..., λN, where
for m > 0.
In fact from the Herglotz representation for n > 0
Carathéodory, C. (1907), "Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen", Math. Ann., 64: 95–115, doi:10.1007/bf01449883, S2CID 116695038
Duren, P. L. (1983), Univalent functions, Grundlehren der Mathematischen Wissenschaften, vol. 259, Springer-Verlag, ISBN0-387-90795-5
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, vol. 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
January 01, 1970
positive, harmonic, function, mathematics, positive, harmonic, function, unit, disc, complex, numbers, characterized, poisson, integral, finite, positive, measure, circle, this, result, herglotz, riesz, representation, theorem, proved, independently, gustav, h. 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 Caratheodory in terms of the positive definiteness of their Taylor coefficients Contents 1 Herglotz Riesz representation theorem for harmonic functions 2 Herglotz Riesz representation theorem for holomorphic functions 3 Caratheodory s positivity criterion for holomorphic functions 4 See also 5 ReferencesHerglotz Riesz representation theorem for harmonic functions editA positive function f on the unit disk with f 0 1 is harmonic if and only if there is a probability measure m on the unit circle such that f r e i 8 0 2 p 1 r 2 1 2 r cos 8 f r 2 d m f displaystyle f re i theta int 0 2 pi 1 r 2 over 1 2r cos theta varphi r 2 d mu varphi nbsp 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 displaystyle f n z f r n z nbsp Then f n r e i 8 1 2 p 0 2 p 1 r 2 1 2 r cos 8 f r 2 f n f d ϕ 0 2 p 1 r 2 1 2 r cos 8 f r 2 d m n f displaystyle 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 nbsp where d m n f 1 2 p f r n e i f d f displaystyle d mu n varphi 1 over 2 pi f r n e i varphi d varphi nbsp 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 m Since rn increases to 1 so that fn z tends to f z the Herglotz formula follows Herglotz Riesz representation theorem for holomorphic functions editA holomorphic function f on the unit disk with f 0 1 has positive real part if and only if there is a probability measure m on the unit circle such that f z 0 2 p 1 e i 8 z 1 e i 8 z d m 8 displaystyle f z int 0 2 pi 1 e i theta z over 1 e i theta z d mu theta nbsp 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 theoremCaratheodory s positivity criterion for holomorphic functions editLet f z 1 a 1 z a 2 z 2 displaystyle f z 1 a 1 z a 2 z 2 cdots nbsp be a holomorphic function on the unit disk Then f z has positive real part on the disk if and only if m n a m n l m l n 0 displaystyle sum m sum n a m n lambda m overline lambda n geq 0 nbsp for any complex numbers l0 l1 lN where a 0 2 a m a m displaystyle a 0 2 a m overline a m nbsp for m gt 0 In fact from the Herglotz representation for n gt 0 a n 2 0 2 p e i n 8 d m 8 displaystyle a n 2 int 0 2 pi e in theta d mu theta nbsp Hence m n a m n l m l n 0 2 p n l n e i n 8 2 d m 8 0 displaystyle 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 nbsp Conversely setting ln zn m 0 n 0 a m n l m l n 2 1 z 2 ℜ f z displaystyle sum m 0 infty sum n 0 infty a m n lambda m overline lambda n 2 1 z 2 Re f z nbsp See also editBochner s theoremReferences editCaratheodory C 1907 Uber den Variabilitatsbereich der Koeffizienten von Potenzreihen die gegebene Werte nicht annehmen Math Ann 64 95 115 doi 10 1007 bf01449883 S2CID 116695038 Duren P L 1983 Univalent functions Grundlehren der Mathematischen Wissenschaften vol 259 Springer Verlag ISBN 0 387 90795 5 Herglotz G 1911 Uber 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 Lehrbucher vol 15 Vandenhoeck amp Ruprecht Riesz F 1911 Sur certains systemes singuliers d equations integrale Ann Sci Ec Norm Super 28 33 62 doi 10 24033 asens 633 Retrieved from https en wikipedia org w index php title Positive harmonic function amp oldid 1117742534, wikipedia, wiki, book, books, library,