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Large Sieve Estimates on Arcs of a Circle

✍ Scribed by Leonid Golinskii; Doron S. Lubinsky; Paul Nevai

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
166 KB
Volume
91
Category
Article
ISSN
0022-314X

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✦ Synopsis

Let 0 [ a < b [ 2p and let D= def {e ih : h ¥ [a, b]}. We show that for generalized (non-negative) polynomials P of degree r and p > 0, we have

where a 1 , a 2 , ..., a m ¥ D, c is an absolute constant (and, thus, it is independent of a, b, p, m, r, P, {a j }) and y is an explicitly determined constant which measures the number of points {a j } in a small interval. This implies large sieve inequalities for generalized (non-negative) trigonometric polynomials of degree r on subintervals of [0, 2p]. The essential feature is the uniformity of the estimate in a and b.

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✍ M. Katchalski; A. Liu 📂 Article 📅 1979 🏛 Elsevier Science 🌐 English ⚖ 963 KB

The object of this paper is to study some intersectional properties of certain families of sets on the circle and on the line. On the circle the families consist of arcs, and on the line they consist of unions of p intervals for some p > 1. The results are related to the classical theorem of Helly a