Trigonometric polynomials with many real zeros (Q1070151)

This is a contribution to the theory of ''incomplete trigonometric polynomials'' \(T_ n\), but mainly for the case when their zeros are not concentrated at just one point, but are distributed in some interval I whose length is not too large. We begin with the simple theorem that if \(\| T_ n\| \leq 1\) and if \(T_ n\) has \(\geq \theta n\), \(0<\theta <2\), zeros at 0, then \(T_ n(t)\) must be small on the interval \(| t| <2\) arcsin (\(\theta\) /2). There are similar but more complicated and more difficult to prove results when \(T_ n\) has \(\geq \theta n\) zeros on I. These results have the following application: If \(T_ n\to f\) a.e., and if \(\| T_ n\|_{\infty}\leq 1\), then f vanishes on a set of the circle T whose measure is controlled by lim sup(N\({}_ n/n)\), where \(N_ n\) is the number of zeros of \(T_ n\) on T. In turn, this has further applications to series of polynomials, to norms of Lagrange operators, and to Hardy classes.