Pólya Class

The Hermite or Pólya class is a set of entire functions satisfying the requirement that if ''E(z)'' is in the class, then:"Polya class theory for Hermite-Biehler functions of finite order"
by Michael Kaltenbäck and Harald Woracek, '' J. London Math. Soc.'' (2) 68.2 (2003), pp. 338–354. .

#''E(z)'' has no zero (root) in the upper half-plane.
#$, E(x+iy), \ge, E(x-iy),$ for ''x'' and ''y'' real and ''y'' positive.
#$, E(x+iy),$ is a non-decreasing function of ''y'' for positive ''y''.

The first condition (no root in the upper half plane) can be derived from the third plus a condition that the function not be identically zero. The second condition is not implied by the third, as demonstrated by the function $\exp(-iz+e^).$ In at least one publication of Louis de Branges, the second condition is replaced by a strict inequality, which modifies some of the properties given below.

Every entire function of Hermite class can be expressed as the limit of a series of polynomials having no zeros in the upper half-plane.

The product of two functions of Hermite class is also of Hermite class, so the class constitutes a monoid under the operation of multiplication of functions.

The class arises from investigations by Georg Pólya in 1913 but some prefer to call it the Hermite class after Charles Hermite.
A de Branges space can be defined on the basis of some "weight function" of Hermite class, but with the additional stipulation that the inequality be strict – that is, $, E(x+iy), >, E(x-iy),$ for positive ''y''. (However, a de Branges space can be defined using a function that is not in the class, such as .)

The Hermite class is a subset of the '' Hermite–Biehler class'', which does not include the third of the above three requirements.

A function with no roots in the upper half plane is of Hermite class if and only if two conditions are met: that the nonzero roots ''z_n'' satisfy

:$\sum_n\frac<\infty$

(with roots counted according to their multiplicity), and that the function can be expressed in the form of a Hadamard product

:$z^m e^\prod_n \left(1-z/z_n\right)\exp(z\operatorname\frac)$

with ''c'' real and non-positive and Im ''b'' non-positive. (The non-negative integer ''m'' will be positive if ''E''(0)=0. Even if the number of roots is infinite, the infinite product is well defined and converges.) From this we can see that if a function of Hermite class has a root at , then $f(z)/(z-w)$ will also be of Hermite class.

Assume is a non-constant polynomial of Hermite class. If its derivative is zero at some point in the upper half-plane, then
:$, f(z), \sim, f(w)+a(z-w)^n,$
near for some complex number and some integer greater than 1. But this would imply that $, f(x+iy),$ decreases with somewhere in any neighborhood of , which cannot be the case. So the derivative is a polynomial with no root in the upper half-plane, that is, of Hermite class. Since a non-constant function of Hermite class is the limit of a sequence of such polynomials, its derivative will be of Hermite class as well.

Louis de Branges showed a connexion between functions of Hermite class and analytic functions whose imaginary part is non-negative in the upper half-plane (UHP), often called Nevanlinna functions. If a function ''E''(''z'') is of Hermite-Biehler class and ''E''(0) = 1, we can take the logarithm of ''E'' in such a way that it is analytic in the UHP and such that log(''E''(0)) = 0. Then ''E''(''z'') is of Hermite class if and only if

:$\text\fracz\ge 0$

(in the UHP).Section 14 of the book by de Branges, or

Laguerre–Pólya class

A smaller class of entire functions is the Laguerre–Pólya class, which consists of those functions which are locally the limit of a series of polynomials whose roots are all real. Any function of Laguerre–Pólya class is also of Hermite class. Some examples are $\sin(z), \cos(z), \exp(z), \text\exp(-z^2).$

Examples

From the Hadamard form it is easy to create examples of functions of Hermite class. Some examples are:
*A non-zero constant.
*$z$
*Polynomials having no roots in the upper half plane, such as $z+i$
*$\exp(-piz)$ if and only if Re(''p'') is non-negative
*$\exp(-pz^2)$ if and only if ''p'' is a non-negative real number
*any function of Laguerre-Pólya class: $\sin(z), \cos(z), \exp(z), \exp(-z), \exp(-z^2).$
*A product of functions of Hermite class

References

{{DEFAULTSORT:Polya class
Analytic functions