Questions tagged [lacunary-series]

A lacunary function, also known as a lacunary series, is an analytic function that cannot be analytically continued anywhere outside the radius of convergence within which it is defined by a power series. The word lacunary is derived from lacuna, meaning gap, or vacancy.

The first known examples of lacunary functions involved Taylor series with large gaps, or lacunae, between the non-zero coefficients of their expansions. More recent investigations have also focused attention on Fourier series with similar gaps between non-zero coefficients. There is a slight ambiguity in the modern usage of the term lacunary series, which may refer to either Taylor series or Fourier series.

Let $ a\in \mathbb {Z} \cap \left[2,\infty \right)$. Consider the following function defined by a simple power series:

$$ f(z)=\sum _{n=0}^{\infty} z^{a^{n}}= z+z^{a}+z^{a^{2}}+z^{a^{3}}+z^{a^{4}}+\cdots $$

The power series converges uniformly on any open domain $|z| < 1$. This can be proved by comparing f with the geometric series, which is absolutely convergent when $|z| < 1$. So $f$ is analytic on the open unit disk. Nevertheless, $f$ has a singularity at every point on the unit circle, and cannot be analytically continued outside of the open unit disk, as the following argument demonstrates.

Clearly $f$ has a singularity at $z = 1$, because

$$ f(1)=1+1+1+\cdots $$

is a divergent series. But if $z$ is allowed to be non-real, problems arise, since

$$ f\left(z^{a}\right)=f(z)-z\qquad f\left(z^{a^{2}}\right)=f(z^{a})-z^{a}\qquad f\left(z^{a^{3}}\right)=f\left(z^{a^{2}}\right)-z^{a^{2}}\qquad \cdots \qquad f\left(z^{a^{n+1}}\right)=f\left(z^{a^{n}}\right)-z^{a^{n}} $$

we can see that $f$ has a singularity at a point $z$ when $z^a = 1$, and also when $z^{a^2} = 1$. By the induction suggested by the above equations, f must have a singularity at each of the $a^n$-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 $f$.

Source: Wikipedia