An interference problem for exponentials (Q1824059)

The problem (posed by H. L. Montgomery) is to determine the supremum C(T) of \[ \int^{T}_{-T}| B(t)|^ 2 dt/\int^{T}_{-T}| A(t)|^ 2dt \] over \(A(t)=\sum^{N}_{k=1}a_ k \exp (i\lambda_ kt)\), \(a_ k>0\), and \(B(t)=\sum^{N}_{k=1}b_ k \exp (i\lambda_ kt)\), where \(| b_ k| \leq a_ k\). This paper discusses several aspects of this and related problems. In particular, the author obtains both upper and lower bounds for C(T). The methods are both ingenious and powerful, but too complicated to be described in detail here. An interesting application is the (best possible) inequality \[ (2T)^{- 1}\int^{T}_{-T}| \sum^{\infty}_{k=-\infty}a_ k \exp (i\lambda_ kt)|^ 2 dt\geq \sum a^ 2_ k, \] where \(a_ k\geq 0\).