Hermite class

The Hermite or Pólya class is a set of entire functions satisfying the requirement that if E(z) is in the class, then:


 * 1) 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 $$\exp(-iz+e^{iz}).$$ 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 $exp(z^{2}−iz)$.)

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 zn satisfy


 * $$\sum_n\frac{1-\operatorname{Im} z_n}{|z_n|^2}<\infty$$

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


 * $$z^m e^{a+bz+cz^2}\prod_n \left(1-z/z_n\right)\exp(z\operatorname{Re}\frac{1}{z_n})$$

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 $f(z)$ of Hermite class has a root at $w$, then $$f(z)/(z-w)$$ will also be of Hermite class.

Assume $f(z)$ is a non-constant polynomial of Hermite class. If its derivative is zero at some point $w$ in the upper half-plane, then
 * $$|f(z)|\sim|f(w)+a(z-w)^n|$$

near $w$ for some complex number $a$ and some integer $n$ greater than 1. But this would imply that $$|f(x+iy)|$$ decreases with $y$ somewhere in any neighborhood of $w$, 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


 * $$\text{Im}\frac{-\log(E(z))}z\ge 0$$

(in the UHP).

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 $$\sin(z), \cos(z), \exp(z), \text{ and }\exp(-z^2).$$

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