Lacunary function

The first known examples of lacunary functions involved Taylor series with large gaps, or lacunae, between the non-zero coefficients of their expansions.

By the induction suggested by the above equations, f must have a singularity at each of the an-th roots of unity for all natural numbers n. The set of all such points is dense on the unit circle, hence by continuous extension every point on the unit circle must be a singularity of f.[1] Evidently the argument advanced in the simple example shows that certain series can be constructed to define lacunary functions.

What is not so evident is that the gaps between the powers of z can expand much more slowly, and the resulting series will still define a lacunary function.

In this context, attention has been focused on criteria sufficient to guarantee convergence of the trigonometric series almost everywhere (that is, for almost every value of the angle θ and of the distortion factor ω).

The geometric series itself defines an analytic function that converges everywhere on the closed unit disk except when z = 1, where g(z) has a simple pole.

From this perspective, then, mathematicians who investigate lacunary series are asking the question: How much does the geometric series have to be distorted – by chopping big sections out, and by introducing coefficients ak ≠ 1 – before the resulting mathematical object is transformed from a nice smooth meromorphic function into something that exhibits a primitive form of chaotic behavior?