Treffer: Some remarks on sequences having a correlation

Let \({\mathcal S}\) be the space introduced by N. Wiener consisting of all complex sequences \(a=(a_ 0,a_ 1,...)\) for which for every non- negative integer k, \(n^{-1}(\bar a_ 0a_ k+\bar a_ 1a_{k+1}+...+\bar a_ na_{n+k})\) has a limit as \(n\to \infty.\) Let \(\gamma_ a\) be the correlation function of \(a\in {\mathcal S},\) i.e., the function which takes the value \(\gamma_ a(k)\) for each k given by the above limit. The correlation function of each sequence in \({\mathcal S}\) is the Fourier transform of a unique non-negative Borel measure on the unit circle. If this measure is Lebesgue measure, we shall say that a has Lebesgue spectrum. The author constructs a,b in \({\mathcal S}\) with Lebesgue spectrum for which \(n^{-1}(\bar a_ 0b_ 0+\bar a_ 1b_ 1+...+\bar a_ nb_ n)\) has no limit.