Large Sieve Estimates on Arcs of a Circle

Abstract

AbstractLet 0⩽α<β⩽2π and let Δ=def{eiθ:θ∈[α, β]}. We show that for generalized (non–negative) polynomials P of degree r and p>0, we have∑j=1m|P(aj)|p|aj−eiα||aj−eiβ|+β−αpr+121/2⩽cτ(pr+1)∫βα|P(eiθ)|pdθ, where a1, a2, …, am∈Δ, c is an absolute constant (and, thus, it is independent of α, β, p, m, r, P, {aj}) and τ is an explicitly determined constant which measures the number of points {aj} 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 uniformity of the estimate in α and β