Conclusions

We have shown that there is a close and deep relationship between real Fourier series and analytic functions in the unit disk centered at the origin of the complex plane. In fact, there is a one-to-one correspondence between pairs of FC Fourier series and the Taylor series of inner analytic functions, so long as in both cases one has convergence at least on a single point of the unit circle. This allows one to use the powerful and extremely well-known machinery of complex analysis to deal with trigonometric series in general, and Fourier series in particular.

One rather unexpected result of this interaction is the fact that the well-known formulas for the Fourier coefficients are in fact a consequence of the structure of the complex analysis of inner analytic functions. Furthermore, so are all the other basic elements of Fourier theory, also in a somewhat unexpected way. We may therefore conclude, from what we have shown here, that the following rather remarkable statement is true:

$\parbox{\textwidth}{\bf The whole structure of the Fourier basis and... ... of the scalar product in the space of real functions generated by the basis.}$

One does not usually associate very weakly convergent Fourier series, that is, series which are not necessarily absolutely and uniformly convergent, with analytic functions, but rather with at most piecewise continuous and differentiable functions that can be non-differentiable or even discontinuous at some points. Therefore, it comes as a bit of a surprise that all such series, if convergent, have limiting functions that are also limits of inner analytic functions on the open unit disk when one approaches the boundary of this maximum disk of convergence of the corresponding Taylor series around the origin. This implies that, so long as there is at most a finite number of sufficiently soft singularities on the unit circle the series in fact converge to piecewise segments of $C^{\infty}$ functions.

Furthermore, the relation with analytic functions in the unit disk allows one to recover the functions that generated a Fourier series from the coefficients of the series, even if the Fourier series itself does not converge. In principle, this can be accomplished by taking limits of the corresponding inner analytic functions from within the unit disk to the unit circle in the complex plane. This helps to give to the Fourier coefficients a definite meaning even when the corresponding series are divergent. Usually, this meaning is thought of in terms of the idea that the coefficients of the series still identify the original function, even if the series does not converge. Now we are able to present a more powerful and general way to recover the function from the coefficients in such cases.

Note that any function $w(z)$ which is analytic on the open unit disk and satisfies the other conditions defining it as an inner analytic function still corresponds to a pair of FC Fourier series at the unit circle, even if these series are divergent everywhere over the unit circle. In fact, to accomplish this correspondence it would be enough to establish that the open unit disk is the maximum disk of convergence of the Taylor series of the inner analytic function $w(z)$, for example using the ratio test. This seems to indicate that the analytic-function structure on the unit disk is the more fundamental one. The fact that we may use the analytic structure to recover the function $f(\theta)$ from the Fourier coefficients, even when the Fourier series is everywhere divergent, seems to indicate the same thing.

The analysis of convergence of a given complete Fourier series will require, in general, the separate discussion of the convergence of the cosine and sine sub-series, and thus will involve two pairs of DP Fourier series and two inner analytic functions $w(z)$. Whether or not each pair of FC Fourier series converge to the corresponding functions depends on whether or not the Taylor series of the analytic function $w(z)$ converges on the unit circle. This is always the case if $w(z)$ is analytic over the whole unit circle, but otherwise its Taylor series may or may not be convergent at some or at all points on that circle. However, when and where $S_{z}$ does converge on the unit circle, that convergence implies the convergence of $S_{v}$ and therefore of the two corresponding FC Fourier series.

Since the convergence of the DP Fourier series is determined by the convergence of the power series $S_{z}$ and $S_{v}$, it also follows from our results here that the convergence or divergence of all these series is ruled by the singularity structure of the inner analytic functions $w(z)$. In the two extreme cases examined here, it is the existence of absence of singularities within the unit disk that determines everything. In the case of strong divergence, the existence of a single singularity within the open unit disk suffices to determine the divergence of $S_{v}$ over the whole unit circle, and thus the divergence of the DP Fourier series almost everywhere over the periodic interval. In the case of strong convergence, the absence of singularities over the whole closed unit disk suffices to establish the strong convergence of $S_{v}$ over the whole unit circle, and thus the strong convergence of the DP Fourier series everywhere over the periodic interval. The remaining case is that in which there are singularities only on the unit circle, and not within the open unit disk. This is the case in which that disk is the maximum disk of convergence of the series $S_{z}$. This is the more complex case, and will be examined in the follow-up paper [1].

It is conceivable that the relation of Fourier theory with complex analysis presented here can be used for other ends, possibly more general and abstract. For example, if one is looking for a necessary and/or sufficient condition for the convergence of Fourier series, one may consider trying to establish such a necessary and/or sufficient condition for the convergence of Taylor series of inner analytic functions on the unit circle, including the cases in which the unit circle is the rim of the maximum disk of convergence of the series. Also, as we have show here for the simple case of the Dirac delta ``function'', it is possible that the distributions associated to such singular objects can be discussed in the complex plane, by means of the use of their Fourier representations.

Since complex analysis and analytic functions constitute such a powerful tool, with so many applications in almost all areas of mathematics and physics, it is to be hoped that other applications of the ideas explored here will in due time present themselves. It should be noted that the relationship with analytic functions and Taylor series constitutes a new way to present the subject of Fourier series, that in fact might become a rather simple and straightforward way to teach the subject.