We introduce an elementary argument to bound the $\textrm{ BMO}$ seminorm of Fourier series with gaps giving in particular a sufficient condition for them to be in this space. Using finer techniques we carry out a detailed study of the series $\sum n^{ -1}e^{ 2\pi i n^2 x}$ providing some insight into how much this $\text{ BMO}$ Fourier series differs from defining an $L^\infty$ function.
Categories: math.CA