A Modified Ratio Test

Let us suppose that the $S_{z}$ series satisfies the ratio convergence test at a point on the unit circle. That immediately implies, of course, that the series converges strongly to an inner analytic function strictly inside the unit disk. However, one can state more than just that, because the test actually implies that the corresponding inner analytic function is also analytic everywhere over the unit circle.

As demonstrated in Section B.2 of Appendix B, if $S_{z}$ satisfies the ratio test at a point on the unit circle then, due to the basic convergence theorem, the maximum disk of convergence of the series $S_{z}$ extends beyond the unit circle, and contains it. It follows therefore that the $S_{z}$ series converges strongly to an analytic function on the whole closed unit disk. It now follows once again that the $S_{v}$ series and the corresponding DP Fourier series $S_{\rm c}$ and $S_{\rm s}$ converge strongly to $C^{\infty}$ functions over the whole unit circle, and hence everywhere over the whole periodic interval.

This constitutes in fact a modified ratio test that can be used for arbitrary DP Fourier series. Given any cosine or sine series, if the coefficients of the series satisfy the ratio test (and we mean here the coefficients $a_{k}$, not the terms of the series), then the series is strongly convergent to a $C^{\infty}$ function on the whole periodic interval. The same is valid for its FC Fourier series, of course. Note that, since the test is to be applied to the coefficients, in this case it is not necessary to test the two FC series independently.