A Strong-Convergence Test

The situation in the strong convergence case suggests the following strong convergence test for an arbitrary DP Fourier series. Given for example a cosine series, and the corresponding FC series, one may write down the pair of extended series

$$\begin{eqnarray*} S'_{\rm c} & = & \sum_{k=1}^{\infty} a_{k}\rho^{k}\cos(k\t... ...{\rm c} & = & \sum_{k=1}^{\infty} a_{k}\rho^{k}\sin(k\theta), \end{eqnarray*}$$

where $\rho\geq 0$ is real. If it can be established that there is a single point, with $\rho>1$ and any value of $\theta$, such that these series are both convergent at that point, then the original cosine series converges strongly to a $C^{\infty}$ function over its whole domain, that is, the whole periodic interval. The values $\theta=0$ and $\theta=\pm\pi$ are sensible choices for the test, because the sine series is always convergent for these values, so that only the cosine series has to be actually tested. The same test holds if we start with a sine series, of course. Additionally, the corresponding FC series are also strongly convergent everywhere on the periodic interval to the respective FC functions.

This same test can also be used to determine whether the given series is strongly divergent, if we revert the condition on $\rho$ and use the test with $\rho<1$, looking now for a point of divergence. In this case it is enough that any one of the two DP trigonometric series of the FC pair diverge at a single point in the interior of the unit disk to ensure that both diverge almost everywhere over the unit circle.