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Abstract <p> We study a problem concerning the exact calculation of the limit of a special sequence of trigonometric functions depending on a real parameter. The problem has recently arisen in the theory of functional-differential operators with affine transformations of the argument, and its complete solution is unknown. By a new elementary method, we find the exact value of the limit for the case in which the parameter is chosen from a certain family of quadratic irrationalities that are Pisot numbers. Estimates are given for the rate of convergence of the corresponding sequence to its limit. For individual parameter values not included in the main family, previously known estimates on the value of the desired limit are improved. </p>

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Asymptotic Behavior of “Long” Products of Sines and the Pisot Numbers

  • 期刊论文

  • 发表日期：

    2025年2月1日

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  • E. D. Alferova,
  • V. E. Podol’skii,
  • V. B. Sherstyukov

摘要

Abstract

We study a problem concerning the exact calculation of the limit of a special sequence of trigonometric functions depending on a real parameter. The problem has recently arisen in the theory of functional-differential operators with affine transformations of the argument, and its complete solution is unknown. By a new elementary method, we find the exact value of the limit for the case in which the parameter is chosen from a certain family of quadratic irrationalities that are Pisot numbers. Estimates are given for the rate of convergence of the corresponding sequence to its limit. For individual parameter values not included in the main family, previously known estimates on the value of the desired limit are improved.