Laguerre–Pólya class

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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:

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

  • The roots are all real.
  • The nonzero zeros zn satisfy
\sum_n\frac{1}{|z_n|^2} converges, with zeros counted according to their multiplicity)
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[edit]

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[edit]

  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.