Monotonic Series

The main remaining problem in our classification of convergence modes in terms of the dominant singularities is to show that if those singularities are borderline hard ones then the $S_{v}$ series is still convergent almost everywhere. While we are not able to prove this in general, even within the realm of the regular integral-differential chains, there are some rather large classes of DP Fourier series for which it is possible to prove the convergence almost everywhere. These series satisfy all the conditions imposed in the previous section, which means that they belong to regular integral-differential chains. We will also be able to establish the existence, location and character of the dominant singularities of the corresponding inner analytic functions $w(z)$.

We will call these series monotonic series, which refers to the fact that they have coefficients $a_{k}$ that behave monotonically with $k$. Other series can also be built from these monotonic series by means of finite linear combinations, which will share their properties regarding convergence and dominant singularities, but which are not themselves monotonic. We will call these extended monotonic series. The method we will use to establish proof of convergence will lead to the concept of singularity factorization, which we will later generalize. This is a method for evaluating the series which is algorithmically useful, and can be used safely in very general circumstances.