A description of the spectrum of Sergeev frequencies of zeros of linear difference equations

ABSTRACT

We prove that the spectra (sets of values) of the upper and lower Sergeev frequencies of zeros of a linear difference equation of order higher than one are Suslin sets of the interval $[0,1].$ Moreover, we prove the inverse theorem for upper frequencies of zeros of second-order equations under the additional assumption that 0 belongs to the spectrum.

KEYWORDS:

• Linear difference equations
• Sergeev frequencies
• numerical characteristics
• asymptotic behaviour

AMS:

• 39A06
• 39A21
• 34D08
• 34D05

Acknowledgements

On the Belarusian side, the work was carried out within the framework of the Special Project of Fundamental and Applied Scientific Research of the National Academy of Sciences of Belarus “Investigation of the properties of spectra of discrete systems under perturbations of their coefficients.”

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research presented here by the second and the fourth author was done as part of the project funded by the National Science Centre in Poland granted according to decision DEC-2017/25/B/ST7/02888.