Large Sieve Estimates on Arcs of a Circle

摘要

Let 0 ≤ α < β ≤ 2π and let Δ=def {e~(iθ):θ ∈ [α,β]}. We show that for generalized (non-negative) polynomials P of degree r and p > 0, we have ∑ from j=1 to m of |P(a_j)|~p(|a_j-e~(iα)||a_J-e~(iβ)|+(β-α/pr+1)~2)~(1/2) ≤ cτ(pr+1)∫_α~β|P(e~(iθ)|~pdθ, where a_1, a_2, ..., a_m ∈ △, c is an absolute constant (and, thus, it is independent of α, β, p, m, r, P, {a_j} and τ 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 [0,2π]. The essential feature is the uniformly of the estimate in α and β.