Laguerre–Pólya class

The Laguerre–Pólya class is the class of entire functions consisting of those functions which are locally the limit of a series of polynomials whose roots are all real. ^[1] Any function of Laguerre–Pólya class is also of Pólya class.

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

Some properties of a function $E(z)$ in the Laguerre–Pólya class are:

• All roots are real.
• $|E(x+iy)|=|E(x-iy)|$ for x and y real.
• $|E(x+iy)|$ is a non-decreasing function of y for positive y.

A function is of Laguerre–Pólya class if and only if three conditions are met:

• The roots are all real.
• The nonzero zeros z_n satisfy

$\sum_n\frac{1}{|z_n|^2}$ converges, with zeros counted according to their multiplicity)

• The function can be expressed in the form of a Hadamard product

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

with b and c real and c non-positive. (The non-negative integer m will be positive if E(0)=0. Note that if the number of zeros is infinite one may have to define how to take the infinite product.)

Examples

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

On the other hand, $\sinh(z), \cosh(z), \text{and } \exp(z^2)$ are not in the Laguerre–Pólya class.

For example,

$\exp(-z^2)=\lim_{n \to \infty}(1-z^2/n)^n.$

Cosine can be done in more than one way. Here is one series of polynomials having all real roots:

$\cos z=\lim_{n \to \infty}((1+iz/n)^n+(1-iz/n)^n)/2$

And here is another:

$\cos z=\lim_{n \to \infty}\prod_{m=1}^n \left(1-\frac{z^2}{((m-\frac{1}{2})\pi)^2}\right)$

This shows the buildup of the Hadamard product for cosine.

If we replace z² with z, we have another function in the class:

$\cos \sqrt z=\lim_{n \to \infty}\prod_{m=1}^n \left(1-\frac z{((m-\frac{1}{2})\pi)^2}\right)$

Another example is the reciprocal gamma function 1/Γ(z). It is the limit of polynomials as follows:

$1/\Gamma(z)=\lim_{n \to \infty}\frac 1{n!}(1-(\ln n)z/n)^n\prod_{m=0}^n(z+m).$

References

1. "Approximation by entire functions belonging to the Laguerre–Pólya class" by D. Dryanov and Q. I. Rahman, Methods and Applications of Analysis" 6 (1) 1999, pp. 21–38.