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Positive harmonic function
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<h2 id="herglotz-riesz-representation-theorem-for-harmonic-functions">Herglotz-Riesz representation theorem for harmonic functions</h2> <p>A positive function <i>f</i> on the unit disk with <i>f</i>(0) = 1 is harmonic if and only if there is a <a href="/facts/Probability_measure/8BjhgoPR">probability measure</a> μ on the unit circle such that </p> f ( r e i θ ) = ∫ 0 2 π 1 − r 2 1 − 2 r cos ( θ − φ ) + r 2 d μ ( φ ) . {\displaystyle f(re^{i\theta })=\int _{0}^{2\pi }{1-r^{2} \over 1-2r\cos(\theta -\varphi )+r^{2}}\,d\mu (\varphi ).} <p>The formula clearly defines a positive harmonic function with <i>f</i>(0) = 1. </p><p>Conversely if <i>f</i> is positive and harmonic and <i>r</i><i>n</i> increases to 1, define </p> f n ( z ) = f ( r n z ) . {\displaystyle f_{n}(z)=f(r_{n}z).\,} <p>Then </p> f n ( r e i θ ) = 1 2 π ∫ 0 2 π 1 − r 2 1 − 2 r cos ( θ − φ ) + r 2 f n ( φ ) d ϕ = ∫ 0 2 π 1 − r 2 1 − 2 r cos ( θ − φ ) + r 2 d μ n ( φ ) {\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 )} <p>where </p> d μ n ( φ ) = 1 2 π f ( r n e i φ ) d φ {\displaystyle d\mu _{n}(\varphi )={1 \over 2\pi }f(r_{n}e^{i\varphi })\,d\varphi } <p>is a probability measure. </p><p>By a compactness argument (or equivalently in this case <a href="/facts/Helly%2527s_selection_theorem/TYBsKs5p">Helly's selection theorem</a> for <a href="/facts/Stieltjes_integral/C08wyfWn">Stieltjes integrals</a>), a subsequence of these probability measures has a weak limit which is also a probability measure μ. </p><p>Since <i>r</i><i>n</i> increases to 1, so that <i>f</i><i>n</i>(<i>z</i>) tends to <i>f</i>(<i>z</i>), the Herglotz formula follows. </p> <h2 id="herglotz-riesz-representation-theorem-for-holomorphic-functions">Herglotz-Riesz representation theorem for holomorphic functions</h2> <p>A holomorphic function <i>f</i> on the unit disk with <i>f</i>(0) = 1 has positive real part if and only if there is a probability measure μ on the unit circle such that </p> f ( z ) = ∫ 0 2 π 1 + e − i θ z 1 − e − i θ z d μ ( θ ) . {\displaystyle f(z)=\int _{0}^{2\pi }{1+e^{-i\theta }z \over 1-e^{-i\theta }z}\,d\mu (\theta ).} <p>This follows from the previous theorem because: </p> <ul><li>the Poisson kernel is the real part of the integrand above</li> <li>the real part of a holomorphic function is harmonic and determines the holomorphic function up to addition of a scalar</li> <li>the above formula defines a holomorphic function, the real part of which is given by the previous theorem</li></ul> <h2 id="carathodorys-positivity-criterion-for-holomorphic-functions">Carathéodory's positivity criterion for holomorphic functions</h2> <p>Let </p> f ( z ) = 1 + b 1 z + b 2 z 2 + ⋯ {\displaystyle f(z)=1+b_{1}z+b_{2}z^{2}+\cdots } <p>be a holomorphic function on the unit disk. Then <i>f</i>(<i>z</i>) has positive real part on the disk if and only if </p> ∑ m ∑ n a m − n λ m λ n ¯ ≥ 0 {\displaystyle \sum _{m}\sum _{n}a_{m-n}\lambda _{m}{\overline {\lambda _{n}}}\geq 0} <p>for any complex numbers λ0, λ1, ..., λ<i>N</i>, where </p> a 0 = 2 , a − m = a m ¯ {\displaystyle a_{0}=2,\,\,\,a_{-m}={\overline {a_{m}}}} <p>for <i>m</i> > 0. </p><p>In fact from the Herglotz representation for <i>n</i> > 0 </p> a n = 2 ∫ 0 2 π e − i n θ d μ ( θ ) . {\displaystyle a_{n}=2\int _{0}^{2\pi }e^{-in\theta }\,d\mu (\theta ).} <p>Hence </p> ∑ m ∑ n a m − n λ m λ n ¯ = ∫ 0 2 π | ∑ n λ n e − i n θ | 2 d μ ( θ ) ≥ 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.} <p>Conversely, setting λ<i>n</i> = <i>z</i><i>n</i>, </p> ∑ m = 0 ∞ ∑ n = 0 ∞ a m − n λ m λ 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).} <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Bochner%2527s_theorem/uQ97G6pP">Bochner's theorem</a></li></ul> <ul><li>Carathéodory, C. (1907), <a href="https://zenodo.org/record/1428260">"Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen"</a>, <i>Math. Ann.</i>, 64: 95–115, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fbf01449883">10.1007/bf01449883</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:116695038">116695038</a></li> <li>Duren, P. L. (1983), <i>Univalent functions</i>, Grundlehren der Mathematischen Wissenschaften, vol. 259, Springer-Verlag, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-90795-5</li> <li>Herglotz, G. (1911), "Über Potenzreihen mit positivem, reellen Teil im Einheitskreis", <i>Ber. Verh. Sachs. Akad. Wiss. Leipzig</i>, 63: 501–511</li> <li><a href="/facts/Christian_Pommerenke/7xpHq182">Pommerenke, C.</a> (1975), <i>Univalent functions, with a chapter on quadratic differentials by Gerd Jensen</i>, Studia Mathematica/Mathematische Lehrbücher, vol. 15, Vandenhoeck & Ruprecht</li> <li>Riesz, F. (1911), "Sur certains systèmes singuliers d'équations intégrale", <i>Ann. Sci. Éc. Norm. Supér.</i>, 28: 33–62, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.24033%2Fasens.633">10.24033/asens.633</a></li></ul>