Strong Convergence

The simplest case of convergence of the DP Fourier series is that in which there is strong convergence to a $C^{\infty}$ function. Let us assume that we have a pair of FC Fourier series and use their coefficients to construct the corresponding $S_{v}$ and $S_{z}$ series. From the basic convergence theorem it follows that, if $S_{z}$ converges at a single point $z_{1}$ strictly outside the closed unit disk, then it is strongly convergent over that whole disk. This means that $w(z)$ is analytic everywhere on the unit circle, and therefore that it is $C^{\infty}$ there. In fact, it means that the maximum disk of convergence of the $S_{z}$ series is larger than the closed unit disk, and contains it. In this case the inner analytic function $w(z)$ has no singularities at all over the whole closed unit disk.

Since the series $S_{v}$ is a restriction of $S_{z}$ to the unit circle, it then follows that $S_{v}$ and the corresponding DP Fourier series are strongly convergent to $C^{\infty}$ functions over the whole unit circle, and hence everywhere over the whole periodic interval. In fact, since this is true for the power series $S_{z}$, in this case the DP Fourier series can be differentiated term-by-term any number of times, and this operation will always result in other equally convergent series. This is the strongest form of convergence one can hope for. It is interesting to note, in passing, that Fourier series displaying this type of convergence appear in the solutions of essentially all boundary value problems involving the diffusion equation in Cartesian coordinates.

Note that in this strong convergence case there is no need to make the distinction between trigonometric series and Fourier series. This is so because, if a trigonometric series converges almost everywhere on the unit circle to a function $f(\theta)$, then the orthogonality of the Fourier basis formed by the sets of cosine and sine functions suffices to show that the coefficients of the series are given in terms of the usual integrals involving $f(\theta)$. Therefore, under these circumstances every trigonometric series is a Fourier series.