Laguerre–Pólya class explained

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 (1)/(|z ^{2} _{n|})} 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), and \exp( - z ^{2).}

On the other hand,

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

For example,

\exp( - z ^{2 ) = \lim} _{n \to inf ty} ( 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} (( 1+ iz/n) ^{n+(1-iz/n)} ^{n)/2} And here is another:\cos z = \lim _{n} \prod ^{n} _{m =1} \left( 1- (z ^{2})/(((m - (1)/()\pi) ^{2} 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} \prod ^{n} _{m =1} \left( 1- (z{((m - (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} (1 {n!}( 1- ( ln)/(n)z/n)) ^{n(z + m).} ^{m =0}

Notes and References

1. http://www.intlpress.com/MAA/p/1999/6_1/MAA-6-1-021-038.pdf "Approximation by entire functions belonging to the Laguerre–Pólya class"