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][2]

1. E(z) has no zero (root) in the upper half-plane.
2. ${\displaystyle |E(x+iy)|\geq |E(x-iy)|}$ for x and y real and y positive.
3. ${\displaystyle |E(x+iy)|}$ 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 ${\displaystyle \exp(-iz+e^{iz}).}$ 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.[3]

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.[4]

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[5] but some prefer to call it the Hermite class after Charles Hermite.[6] 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, ${\displaystyle |E(x+iy)|>|E(x-iy)|}$ for positive y. (However, a de Branges space can be defined using a function that is not in the class, such as exp(z²−iz).)

The Hermite class is a subset of the Hermite–Biehler class, which does not include the third of the above three requirements.[2]

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

${\displaystyle \sum _{n}{\frac {1-\operatorname {Im} z_{n}}{|z_{n}|^{2}}}<\infty }$

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

${\displaystyle z^{m}e^{a+bz+cz^{2}}\prod _{n}\left(1-z/z_{n}\right)\exp(z\operatorname {Re} {\frac {1}{z_{n}}})}$

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.[7]) From this we can see that if a function f(z) of Hermite class has a root at w, then ${\displaystyle f(z)/(z-w)}$ 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

${\displaystyle |f(z)|\sim |f(w)+a(z-w)^{n}|}$

near w for some complex number a and some integer n greater than 1. But this would imply that ${\displaystyle |f(x+iy)|}$ 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.[8]

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

${\displaystyle {\text{Im}}{\frac {-\log(E(z))}{z}}\geq 0}$

(in the UHP).[9]