Weak Convergence, Classification and

Factorization of Singularities

The convergence of DP Fourier series which are neither strongly
convergent nor strongly divergent is discussed in terms of the Taylor
series of the corresponding inner analytic functions. These are the
cases in which the maximum disk of convergence of the Taylor series of
the inner analytic function is the open unit disk. An essentially
complete classification, in terms of the singularity structure of the
corresponding inner analytic functions, of the modes of convergence of a
large class of DP Fourier series, is established. Given a weakly
convergent Fourier series of a DP real function, it is shown how to
generate from it other expressions involving trigonometric series, that
converge to that same function, but with much better convergence
characteristics. This is done by a procedure of factoring out the
singularities of the corresponding inner analytic function, and works
even for divergent Fourier series. This can be interpreted as a
resummation technique, which is firmly anchored by the underlying
analytic structure.

- Introduction
- Weak Convergence

- Classification of Singularities
- Convergence and Singularities
- Monotonic Series

- Factorization of Singularities

- Conclusions
- Acknowledgements
- Appendix: Technical Proofs

- Appendix: Examples of Center Series
- A Regular Sine Series with All
- A Regular Sine Series with Odd
- A Regular Sine Series with Even
- A Regular Cosine Series with Odd
- A Singular Cosine Series
- Another Regular Cosine Series with Odd
- Another Regular Sine Series with Odd

- Bibliography