Positive Harmonic Function
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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a positive harmonic function on the
unit disc In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose di ...
in the
complex numbers In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
is characterized as the
Poisson integral In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel can be understood as the deriv ...
of a finite
positive measure In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simi ...
on the circle. This result, the ''Herglotz-Riesz representation theorem'', was proved independently by
Gustav Herglotz Gustav Herglotz (2 February 1881 – 22 March 1953) was a German Bohemian physicist best known for his works on the theory of relativity and seismology. Biography Gustav Ferdinand Joseph Wenzel Herglotz was born in Volary num. 28 to a public not ...
and
Frigyes Riesz Frigyes Riesz ( hu, Riesz Frigyes, , sometimes spelled as Frederic; 22 January 1880 – 28 February 1956) was a HungarianEberhard Zeidler: Nonlinear Functional Analysis and Its Applications: Linear monotone operators. Springer, 199/ref> mathema ...
in 1911. It can be used to give a related formula and characterization for any
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
on the unit disc with positive real part. Such functions had already been characterized in 1907 by
Constantin Carathéodory Constantin Carathéodory ( el, Κωνσταντίνος Καραθεοδωρή, Konstantinos Karatheodori; 13 September 1873 – 2 February 1950) was a Greek mathematician who spent most of his professional career in Germany. He made significant ...
in terms of the
positive definiteness In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positive-definite bilinear form * Positive-definite fun ...
of their
Taylor coefficient In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor seri ...
s.


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 In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more gener ...
μ on the unit circle such that : f(re^)=\int_0^ \, d\mu(\varphi). The formula clearly defines a positive harmonic function with ''f''(0) = 1. Conversely if ''f'' is positive and harmonic and ''r''''n'' increases to 1, define : f_n(z)=f(r_nz). \, Then : f_n(re^) = \int_0^ \, f_n(\varphi)\,d\phi =\int_0^ d\mu_n(\varphi) where : d\mu_n(\varphi)= f(r_n e^)\,d\varphi is a probability measure. By a compactness argument (or equivalently in this case
Helly's selection theorem In mathematics, Helly's selection theorem (also called the ''Helly selection principle'') states that a uniformly bounded sequence of monotone real functions admits a convergent subsequence. In other words, it is a sequential compactness theorem ...
for
Stieltjes integral Thomas Joannes Stieltjes (, 29 December 1856 – 31 December 1894) was a Dutch mathematician. He was a pioneer in the field of moment problems and contributed to the study of continued fractions. The Thomas Stieltjes Institute for Mathematics at ...
s), a subsequence of these probability measures has a weak limit which is also a probability measure μ. Since ''r''''n'' increases to 1, so that ''f''''n''(''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^ \, 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_ \lambda_m\overline \ge 0 for any complex numbers λ0, λ1, ..., λ''N'', where : a_0=2,\,\,\, a_ =\overline for ''m'' > 0. In fact from the Herglotz representation for ''n'' > 0 : a_n =2\int_0^ e^\, d\mu(\theta). Hence :\sum_m\sum_n a_ \lambda_m\overline =\int_0^ \left, \sum_ \lambda_n e^\^2 \, d\mu(\theta) \ge 0. Conversely, setting λ''n'' = ''z''''n'', :\sum_^\infty\sum_^\infty a_ \lambda_m\overline = 2(1-, z, ^2) \,\Re\, f(z).


See also

*
Bochner's theorem In mathematics, Bochner's theorem (named for Salomon Bochner) characterizes the Fourier transform of a positive finite Borel measure on the real line. More generally in harmonic analysis, Bochner's theorem asserts that under Fourier transform a c ...


References

* * * * *{{citation, last=Riesz, first=F., title=Sur certains systèmes singuliers d'équations intégrale, journal=Ann. Sci. Éc. Norm. Supér., volume=28, pages= 33–62, year=1911, doi=10.24033/asens.633, doi-access=free Harmonic analysis Complex analysis Harmonic functions