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

"Approximation by entire functions belonging to the Laguerre–Pólya class" Archived 2008-10-06 at the Wayback Machine by D. Dryanov and Q. I. Rahman, Methods and Applications of Analysis 6 (1) 1999, pp. 21–38.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] "Approximation by entire functions belonging to the Laguerre–Pólya class" Archived 2008-10-06 at the Wayback Machine by D. Dryanov and Q. I. Rahman, Methods and Applications of Analysis 6 (1) 1999, pp. 21–38.