Lacunary series

Summary

A lacunary series is a trigonometric series or a power series whose nonzero terms have frequencies that grow at least geometrically, such as $1, 2, 4, 8, \ldots$. The large gaps make the terms behave almost like independent random variables. As a result, lacunary series have rigid properties. By the Hadamard gap theorem, a power series with such gaps cannot be continued past its circle of convergence. A lacunary Fourier series of a bounded function converges absolutely.