On a problem of G. G. Lorentz regarding the norms of Fourier projections

Abstract:

For any $0 < \alpha < \frac {1}{2}$ we construct a sequence of integers $\left ( {{\mu _1}, \ldots , {\mu _n}, \ldots } \right )$ such that the norms of Fourier projections \[ {F_N} = \sum \limits _1^N {{e^{i{\mu _j}\theta }}} \otimes {e^{i{\mu _{{j^t}}}}}:{C_{\left [ { - \pi ,\pi } \right ]}} \to {C_{\left [ { - \pi ,\pi } \right ]}}\] grow as ${N^\alpha }$ . This answers a question of Prof. G. G. Lorentz.