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] 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.
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.
[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,
(However, a de Branges space can be defined using a function that is not in the class, such as exp(z2−iz).)
[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 zn satisfy (with roots counted according to their multiplicity), and that the function can be expressed in the form of a Hadamard product with c real and non-positive and Im b non-positive.
If its derivative is zero at some point w in the upper half-plane, then near w for some complex number a and some integer n greater than 1.
So the derivative is a polynomial with no root in the upper half-plane, that is, of Hermite class.
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.
[9] 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.
From the Hadamard form it is easy to create examples of functions of Hermite class.