Behavior of a Trigonometric Polynomial with Given Number of Terms on SmallSubsets of the Unit Circumference ('Norm Duplication' Problem)

摘要

This paper can be considered as an appendix to the author's previous paper 'Localestimates of exponential polynomials and their applications to inequalities of the Uncertainty Principle type', where the author obtained some rather sharp estimates for the norm of an exponential polynomial on the interval I (included in) R through its number of terms and its norm on some small measurable subset E (included in) I. The question arose whether one can obtain some good estimates in the case when the measure of E is not small, but, inversely, very close to the measure of I. It turns out that it is rather difficult and one still cannot answer it in the general case. Nevertheless, one particular problem can be solved which seems to be of special interest for applications: the so-called 'norm duplication' problem. Roughly speaking, the question is how close must the measure of E be to that of I to guarantee that the norm of the polynomial on the whole interval I does not exceed more than twice its norm on E. This is discussed in this document.