Supriya Ghosh (Editor)

Pólya class

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

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

Contents

  • E(z) has no zero (root) in the upper half-plane.
  • | E ( x + i y ) | ≥ | E ( x − i y ) | for x and y real and y positive.
  • | E ( x + i y ) | is a non-decreasing function of y for positive y.

    • The first condition (no root in the upper half plane) can be derived from the third plus a condition that the function not be identically zero. The second condition is not implied by the third, as demonstrated by the function exp ⁡ ( − i z + e i z ) . In at least one publication of Louis de Branges, the second condition is replaced by a strict inequality, which modifies some of the properties given below.

    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 arises from investigations by Georg Pólya in 1913. A de Branges space can be defined on the basis of some "weight function" of Pólya class, but with the additional stipulation that the inequality be strict – that is, | E ( x + i y ) | > | E ( x − i y ) | for positive y. (However, a de Branges space can be defined using a function that is not in the class, such as exp(z2−iz).)

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

    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 zn satisfy

    ∑ n 1 − Im ⁡ z n | z n | 2 < ∞

    (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 + b z + c z 2 ∏ n ( 1 − z / z n ) exp ⁡ ( z Re ⁡ 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. Even if the number of roots is infinite, the infinite product is well defined and converges.)

    Louis de Branges showed a connexion between functions of Pólya class and analytic functions whose imaginary part is non-negative in the upper half-plane (UHP), often called Nevanlinna functions. If a function E(z) is of Hermite-Biehler class and E(0) = 1, we can take the logarithm of E in such a way that it is analytic in the UHP and such that log(E(0)) = 0. Then E(z) is of Pólya class if and only if

    Im − log ⁡ ( E ( z ) ) z ≥ 0

    (in the UHP).

    Laguerre–Pólya class

    A smaller 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 ) ,  and  exp ⁡ ( − z 2 ) .

    Examples

    From the Hadamard form it is easy to create examples of functions of Pólya class. Some examples are:

  • A non-zero constant.
  • z
  • Polynomials having no roots in the upper half plane, such as z + i
  • exp ⁡ ( − p i z ) if and only if Re(p) is non-negative
  • exp ⁡ ( − p z 2 ) if and only if p is a non-negative real number
  • any function of Laguerre-Pólya class: sin ⁡ ( z ) , cos ⁡ ( z ) , exp ⁡ ( z ) , exp ⁡ ( − z ) , exp ⁡ ( − z 2 ) .
  • A product of functions of Pólya class

    • References

    Pólya class Wikipedia
    (Text) CC BY-SA