Linear finite difference operators preserving the Laguerre–Pólya class

ABSTRACT

We prove that the difference operator of the following form:

where and are some complex functions and h is a non-zero complex number, preserves the Laguerre–Pólya class of entire functions if and only if, the step h is pure imaginary, , and the function is entire of order at most 2 and belongs to a subclass of the Hermite–Biehler class of entire functions.

KEYWORDS:

• Laguerre–Pólya class
• finite difference operators
• Hermite–Biehler class
• distribution of zeros

AMS Subject Classifications:

• Primary: 30C15
• 30D15
• Secondary: 30D35
• 26C10

Acknowledgements

We are deeply grateful to the referees for their important and valuable remarks.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 The case and can be considered analogously.

2 The case, can be considered analogously.

Additional information

Funding

The second author is Shanghai Oriental Scholar whose research was supported by Russian Science Foundation [grant number 14–11–00022].