Conjugate Pairs of Fourier Series

and Inner Analytic Functions

A correspondence between arbitrary Fourier series and certain analytic
functions on the unit disk of the complex plane is established. The
expression of the Fourier coefficients is derived from the structure of
complex analysis. The orthogonality and completeness relations of the
Fourier basis are derived in the same way. It is shown that the limiting
function of any Fourier series is also the limit to the unit circle of
an analytic function in the open unit disk. An alternative way to
recover the original real functions from the Fourier coefficients, which
works even when the Fourier series are divergent, is thus presented. The
convergence issues are discussed up to a certain point. Other possible
uses of the correspondence established are pointed out.

- Introduction
- Trigonometric Series on the Complex Plane
- Fourier Series on the Complex Plane
- Fourier-Taylor Correspondence
- The Orthogonality Relations
- The Completeness Relation
- Limits from Within
- Some Basic Convergence Results

- Conclusions
- Acknowledgements
- Appendix: Riemann's Function
- Appendix: Technical Proofs

- Appendix: Examples of Limits from Within
- 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

- Bibliography