Weakening the Convergence Hypothesis

Let $f(\theta)$ be a DP real function that has zero average value, with $\theta\in[-\pi,\pi]$. We will assume that $f(\theta)$ is an integrable function, so that its Fourier coefficients $\alpha_{k}$ and $\beta_{k}$ exist, since they are given by

$$\begin{eqnarray*} \alpha_{k} & = & \frac{1}{\pi} \int_{-\pi}^{\pi}d\theta\, ... ...c{1}{\pi} \int_{-\pi}^{\pi}d\theta\, f(\theta) \sin(k\theta). \end{eqnarray*}$$

Note that $\alpha_{0}=0$ because $f(\theta)$ is assumed to be a zero-average function. We will name such coefficients $a_{k}$, for $k=1,2,3,\ldots,\infty$, irrespective of whether $f(\theta)$ is even or odd, and thus of whether it gives origin respectively to a cosine or sine series. Let us assume that the $a_{k}$ coefficients satisfy the following hypothesis: given a sequence $a_{k}$ of coefficients, there exists a positive real function $g(k)$ with the property that

$$\lim_{k\to\infty}g(k) = 1,$$ (1)

and such that for all $k\geq 1$

$$\left\vert\frac{a_{k+1}}{a_{k}}\right\vert \leq g(k).$$ (2)

What this means is that we assume that the ratio of the absolute values of two consecutive coefficients is bounded from above by a function that tends to one in the $k\to\infty$ limit. Note that the $k\to\infty$ limit of the ratio of coefficients itself may not even exist. Note also that, if this condition holds for a given DP real function, then it automatically holds for the corresponding FC real function as well, since both have the same coefficients and the condition is imposed only on these coefficients. Following the development described in [1], given $f(\theta)$ with such properties we may now construct the two DP trigonometric series

$$\begin{eqnarray*} S_{\rm c} & = & \sum_{k=1}^{\infty} a_{k}\cos(k\theta), \\ S_{\rm s} & = & \sum_{k=1}^{\infty} a_{k}\sin(k\theta), \end{eqnarray*}$$

which are the FC series of one another, and then the complex power series

$$\begin{eqnarray*} S_{z} & = & \sum_{k=1}^{\infty} a_{k}\rho^{k} \left[ \co... ...(k\theta) \right] \\ & = & \sum_{k=1}^{\infty} a_{k}z^{k}, \end{eqnarray*}$$

where $z=\rho\exp(\mbox{\boldmath$\imath$}\theta)$, with $\rho\geq 0$, so that $S_{\rm c}$ and $S_{\rm s}$ are respectively the real and imaginary parts of $S_{z}$ for $\rho=1$. Note that the condition expressed by Equations (1) and (2) does not, by itself, imply the convergence of the series $S_{\rm c}$ and $S_{\rm s}$ on the unit circle, since sequences of coefficients $a_{k}$ that do not go to zero as $k\to\infty$ may satisfy it. Therefore, many sequences of coefficients leading to Fourier series that diverge on the unit circle satisfy that condition. If we now use the ratio criterion to analyze the convergence of the power series $S_{z}$ inside the open unit disk, we get

$$\left\vert\frac{a_{k+1}z^{k+1}}{a_{k}z^{k}}\right\vert = \left\vert\frac{a_{k+1}}{a_{k}}\right\vert\rho.$$

Our hypothesis about the ratio of the $a_{k}$ coefficients now leads to the relation, inside the open unit disk,

$$\left\vert\frac{a_{k+1}z^{k+1}}{a_{k}z^{k}}\right\vert \leq \rho g(k).$$

Now, given any value of $\rho<1$, since the limit of $g(k)$ for $k\to\infty$ is one, it follows that the same limit of $\rho g(k)$ is strictly less than one. Therefore, we may conclude that there is a value $k_{m}$ of $k$ such that, if $k>k_{m}$, then $\rho g(k)<1$, so that the ratio is less that one, thus implying that we have

$$\left\vert\frac{a_{k+1}z^{k+1}}{a_{k}z^{k}}\right\vert < 1,$$

for $k>k_{m}$. This implies that the ratio criterion is satisfied and therefore that the series $S_{z}$ converges at such points. Since this is true for any $\rho<1$, we may conclude that the power series $S_{z}$ converges on the open unit disk. Therefore, the series converges to a complex analytic function $w(z)$ there, which is an inner analytic function according to the definition given in [1].

The results established in [1] then imply that $f(\theta)$ can now be obtained almost everywhere over the unit circle as the $\rho\to 1$ limit of the real or imaginary part of $w(z)$, as the case may be. Note that the series $S_{z}$, and hence the series $S_{\rm c}$ and $S_{\rm s}$, may still be divergent over the whole unit circle. This does not affect the recovery of the real function from its Fourier coefficients in the manner just described. We have therefore shown that the correspondence established in [1] holds without the hypothesis that $S_{\rm c}$ and $S_{\rm s}$ be convergent together on at least at one point of the unit circle, so long as the Taylor-Fourier coefficients $a_{k}$ satisfy the weaker condition expressed by Equations (1) and (2).