Pólya class

The Pólya class or Hermite class is a set of entire functions satisfying the requirement that if E(z) is in the class, then:^[1][2]

• 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.^[3]

Every entire function of Pólya 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 Pólya class is also of Pólya class, so the class constitutes a monoid under the operation of multiplication of functions.

The Pólya class arises from investigations by Georg Pólya in 1913.^[5] A de Branges space can be defined on the basis of some "weight function" of Pólya 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(z²−iz).)

The Pólya 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 Pólya class if and only if two conditions are met: that the nonzero roots z_n 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.^[6])

Louis de Branges showed a connexion between functions of Pólya 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 Pólya class if and only if

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

(in the UHP).^[7]

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 Pólya 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 Pólya 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 Pólya class

References

1. Louis de Branges (1968). Hilbert spaces of entire functions. London: Prentice-Hall. ISBN 978-0133889000.
2. "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. DOI: 10.1112/S0024610703004502.
3. Louis de Branges (Jul 1992). "The Convergence of Euler Products" (PDF). Journal of Functional Analysis. doi:10.1016/0022-1236(92)90103-P.
4. Louis de Branges. "A proof of the Riemann Hypothesis" (PDF). p. 6. Archived from the original (PDF) on Nov 9, 2006.
5. G. Polya: "Über Annäherung durch Polynome mit lauter reellen Wurzeln", Rend. Circ. Mat. Palermo 36 (1913), 279-295.
6. Section 7 of the book by de Branges.
7. Section 14 of the book by de Branges, or Louis de Branges (1963). "Some applications of spaces of entire functions". Canadian Journal of Mathematics 15: 563–83. doi:10.4153/CJM-1963-058-1.