Fourier-Taylor Correspondence

Let us now show that there is a complete one-to-one correspondence between arbitrarily given pairs of real FC Fourier series and the inner analytic functions within the open unit disk. To this end, let us imagine that one begins the whole argument of the last section over again, but this time starting with a pair of FC Fourier series. What this means is that there is a zero-average real function $f(\theta)$ defined in the periodic interval such that the coefficients $a_{k}$ are given in terms of that function by the integrals

$$a_{k} = \frac{1}{\pi} \int_{-\pi}^{\pi}d\theta\, f_{\rm c}(\theta) \cos(k\theta)$$

$$= \frac{1}{\pi} \int_{-\pi}^{\pi}d\theta\, f_{\rm s}(\theta) \sin(k\theta),$$ (3)

where we have $f(\theta)=f_{\rm c}(\theta)+f_{\rm s}(\theta)$, with $f_{\rm c}(\theta)$ even on $\theta$ and $f_{\rm s}(\theta)$ odd on $\theta$. The two Fourier series generated by $f_{\rm c}(\theta)$ and $f_{\rm s}(\theta)$ have exactly the same coefficients, and are therefore the FC series of one another. We may therefore consider the corresponding functions to be FC functions of one another as well, even if the series do not converge. We will denote these FC functions by $\bar{f}_{\rm c}(\theta)$ in the case of an original cosine series with coefficients given by $f_{\rm c}(\theta)$, and by $\bar{f}_{\rm s}(\theta)$ in the case of an original sine series with coefficients given by $f_{\rm s}(\theta)$. We have therefore that

$$\begin{eqnarray*} f_{\rm c}(\theta) & = & \bar{f}_{\rm s}(\theta), \\ f_{\rm s}(\theta) & = & \bar{f}_{\rm c}(\theta). \end{eqnarray*}$$

Note that $\bar{f}_{\rm c}(\theta)$ is in fact odd instead of even, while $\bar{f}_{\rm s}(\theta)$ is in fact even instead of odd.

As before, we assume that there is at least one value of $\theta$ for which both series in the FC pair converge. We may then use the coefficients $a_{k}$ to define the inner analytic function $w(z)$, as we did in the previous section, we may identify these coefficients as those of its Taylor series, and therefore these same coefficients turn out to be given by

$$\begin{eqnarray*} a_{k} & = & \frac{\rho^{-k}}{\pi} \int_{-\pi}^{\pi}d\theta... ...nt_{-\pi}^{\pi}d\theta\, f_{\rm s}(\rho,\theta) \sin(k\theta), \end{eqnarray*}$$

where $f_{\rm c}(\rho,\theta)$ and $f_{\rm s}(\rho,\theta)$ are respectively the real and imaginary parts of $w(z)$. Since the coefficients $a_{k}$ are in fact independent of $\rho$ in this last set of expressions, for $0<\rho<1$, and by construction have the same values as those given by the previous set of expressions, in Equation (3), which define them as Fourier coefficients of $f(\theta)$, we see that they are continuous from within as functions of $\rho$ in the limit $\rho\to 1$, which we may then take.

According to the results of the previous section, we conclude therefore that the two FC Fourier series we started with in this section are in fact the DP Fourier series of the functions $f_{\rm c}(\rho,\theta)$ and $f_{\rm s}(\rho,\theta)$ in the $\rho\to 1$ limit. Since the coefficients uniquely identify the function almost everywhere, as discussed in the introduction, we may now identify these two functions in the $\rho\to 1$ limit with the two functions $f_{\rm c}(\theta)$ and $f_{\rm s}(\theta)$ from which the coefficients $a_{k}$ were obtained in the first place, on the unit circle. We conclude that the limits

$$\begin{eqnarray*} f_{\rm c}(\theta) & = & \lim_{\rho\to 1} f_{\rm c}(\rho,\t... ...\rm s}(\theta) & = & \lim_{\rho\to 1} f_{\rm s}(\rho,\theta), \end{eqnarray*}$$

exist and hold almost everywhere over the unit circle. We may consider the $\rho\to 1$ limits of the functions $f_{\rm c}(\rho,\theta)$ and $f_{\rm s}(\rho,\theta)$ to be the maximally smooth members of sets of functions that are zero-measure equivalent and lead to the same set of Fourier coefficients. The functions $f_{\rm c}(\rho,\theta)$ and $f_{\rm s}(\rho,\theta)$, which according to our definitions are the Fourier Conjugate functions of each other, are also known as Harmonic Conjugate functions [4], since they are the real and imaginary parts of an analytic function, and hence are both harmonic functions on the real plane. They can be obtained from one another by the Hilbert transform [5]. In the $\rho\to 1$ limit we may also get functions $f_{\rm c}(\theta)$ and $f_{\rm s}(\theta)$ which are restriction to the unit circle of Harmonic Conjugate functions, if the function $w(z)$ is analytic at the limiting point. But in any case they are Fourier Conjugate to each other.

Up to this point we have established that every given pair of FC Fourier series, obtained from a given pair of FC real functions, such that both converge together on at least one point on the unit circle, corresponds to a specific inner analytic function, whose Taylor series converges on at least that point in the unit circle, and which reproduces the original pair of FC real function almost everywhere when one takes the $\rho\to 1$ limit from within the open unit disk to the unit circle.

Furthermore, we also see that we may work this argument in reverse. In other words, given an arbitrary inner analytic function $w(z)$ whose Taylor series converges on at least one point in the unit circle and which is well-defined almost everywhere over that circle, we may construct from it the DP Fourier series of two related functions. These two DP Fourier series are just the real and imaginary parts of the Taylor series of the analytic function $w(z)$, in the $\rho\to 1$ limit. Finally, since the coefficients $a_{k}$ are continuous functions of $\rho$ from within at the unit disk, and since the $S_{z}$ series converges on at least one point on the unit circle, we may also conclude that the two corresponding FC Fourier series converge together on that point of the unit circle.

This completes the establishment of a one-to-one correspondence between, on the one hand, pairs of real FC Fourier series that converge together on at least a single point of the periodic interval and, on the other hand, inner analytic functions that converge on at least one point of the unit circle and are well-defined almost everywhere over that circle. Note, however, that the analytic side of this correspondence is the more powerful one, because given the Fourier coefficients $a_{k}$ we may be able to define a convergent power series $S_{z}$ and thus an inner analytic function $w(z)$ even if the corresponding Fourier series diverge everywhere on the periodic interval.

This correspondence can be a useful tool, as it may make it easier to determine the convergence or lack thereof of given Fourier series. It may also be used to recover from the coefficients $a_{k}$ the function which generates a given Fourier series, even if that series is divergent. The way to do this is simply to determine the corresponding inner analytic function and then calculate the $\rho\to 1$ limit of the real and imaginary parts of that function. In Appendix C we will give a few simple examples of this type of procedure.