Italo Jose Dejter

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

• 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.

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.

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 was defined by Louis de Branges and arises from investigations by Georg Pólya in 1913. A de Branges space is defined on the basis of some "weight function" of Pólya class.

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

De Branges found that 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

$\sum _n\frac{1-\mathrm{\Im }{z}_n}{|z_n{|}^2}<\mathrm{\infty }$

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

$z^m{e}^{a+bz+c{z}^2}\prod _n(1-z/z_n)\exp (z\mathrm{\Re }\frac{1}{z_n})$

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

Examples

From the Hadamard form it is easy to see some examples of functions of Pólya class:

• Polynomials having no roots in the upper half plane, such as $z+i$
• $exp(-iz)$
• $exp(-z^2)$
• A product of functions of Pólya class

Laguerre–Pólya class

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

References

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