Positive harmonic function

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

(

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 ).}

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).\,}

Then

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\varphi =\int _{0}^{2\pi }{1-r^{2} \over 1-2r\cos(\theta -\varphi )+r^{2}}d\mu _{n}(\varphi )}

where

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 }

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.