Introduction

In this paper we will establish an interesting relation between Fourier series and analytic functions. This leads to an alternative way to deal with Fourier series and to characterize the corresponding real functions. This relation will allow us to discuss the convergence of Fourier series in terms of the convergence of the Taylor series of analytic functions. The convergence issues will be developed up to a certain point, and further developments will be discussed in a follow-up paper [1]. This relation will also give us an alternative way to recover, from the coefficients of the series, the functions that originated them, which works even if the Fourier series are divergent. Perhaps most importantly, it will provide a different and possibly richer point of view for Fourier series and the corresponding real functions.

We will use repeatedly the following very well-known and fundamental theorem of complex analysis, about complex power series [2]. If we consider the general complex power series written around the origin $z=0$,

$$S_{z} = \sum_{k=0}^{\infty} c_{k}z^{k},$$

where $z=x+\mbox{\boldmath$\imath$}y$ is a complex variable and $c_{k}$ are arbitrary complex constants, then the following holds. If $S_{z}$ converges at a point $z_{1}\neq 0$, then it is convergent and absolutely convergent on an open disk centered at $z=0$ with its boundary passing through $z_{1}$. In addition to this, it converges uniformly on any closed set contained within this open disk. We will refer to this state of affairs in what regards convergence as strong convergence, and will refer to this theorem as the basic convergence theorem. Furthermore, the power series converges to an analytic function of which it is the Taylor series around $z=0$.

We will make a conceptual distinction between trigonometric series and Fourier series. An arbitrary real trigonometric series on the real variable $\theta$ with domain in the periodic interval $[-\pi,\pi]$ is given by

$$S = \frac{1}{2}\, \alpha_{0} + \sum_{k=1}^{\infty} \a... ...\cos(k\theta) + \sum_{k=1}^{\infty} \beta_{k}\sin(k\theta),$$

where $\alpha_{k}$ and $\beta_{k}$ are any real numbers. If there is a real function $f(\theta)$ such that the coefficients $\alpha_{k}$ and $\beta_{k}$ are given in terms of that function by the integrals

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

then we call this series the Fourier series of that function [3]. Since the Fourier coefficients are defined by means of integrals, it is clear that one can add to $f(\theta)$ any zero-measure function without modifying them. This means that a convergent Fourier series can only be said to converge almost everywhere to the function which originated it, that is, with the possible exclusion of a zero-measure subset of the domain.

With this limitation, the Fourier basis in the space of real functions on the periodic interval, formed by the constant function, the set of functions $\cos(k\theta)$, with $k=1,\ldots,\infty$, and the set of functions $\sin(k\theta)$, with $k=1,\ldots,\infty$, is complete to generate all sufficiently well-behaved functions in that interval. With the exclusion of the constant function, the remaining basis generates the set of all sufficiently well-behaved zero-average real functions. The remaining basis functions satisfy the orthogonality relations

$$\frac{1}{\pi} \int_{-\pi}^{\pi}d\theta\, \cos(k_{1}\theta) \cos(k_{2}\theta) = \delta_{k_{1}k_{2}},$$

$$\frac{1}{\pi} \int_{-\pi}^{\pi}d\theta\, \sin(k_{1}\theta) \sin(k_{2}\theta) = \delta_{k_{1}k_{2}},$$

$$\frac{1}{\pi} \int_{-\pi}^{\pi}d\theta\, \cos(k_{1}\theta) \sin(k_{2}\theta) = 0,$$ (1)

for $k_{1}\geq 1$ and $k_{2}\geq 1$. In a stricter sense, the good-behavior conditions on the real functions are that they be integrable and that they be such that their Fourier series converge. However, one might consider the set of coefficients themselves to be sufficient to characterize the function that originated them, even if the series does not converge. This makes full sense if there is an alternative way to recover the functions from the coefficients of their series, such as the one we will present in this paper, which is not dependent on the convergence of the series. In this case the condition of integrability suffices. The important point to be kept in mind here is that the set of coefficients determines the function uniquely almost everywhere over the periodic interval.

The parity of the real functions will play an important role in this paper. Any real function defined in a symmetric domain around zero, without any additional hypotheses, can be separated into its even and odd parts. An even function is one that satisfies the condition $f(-\theta)=f(\theta)$, while an odd one satisfies the condition $f(-\theta)=-f(\theta)$. For any real function we can write that $f(\theta)=f_{\rm c}(\theta)+f_{\rm s}(\theta)$, with

$$\begin{eqnarray*} f_{\rm c}(\theta) & = & \frac{f(\theta)+f(-\theta)}{2}, \\ f_{\rm s}(\theta) & = & \frac{f(\theta)-f(-\theta)}{2}, \end{eqnarray*}$$

where $f_{\rm c}(\theta)$ is even and $f_{\rm s}(\theta)$ is odd. The Fourier basis can also be separated into even and odd parts. Since the constant function and the cosines are even, they generate the even parts of the real functions, while the sines, being odd, generate the odd parts. Since the set of cosines and the set of sines are two independent and mutually orthogonal sets of functions, the convergence of the trigonometric series can only be accomplished by the separate convergence of the cosine sub-series and the sine sub-series, which we denote by

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

so that $S=\alpha_{0}/2+S_{\rm c}+S_{\rm s}$. For simplicity, in this paper we will consider only series in which $\alpha_{0}=0$, which correspond to functions $f(\theta)$ that have zero average over the periodic interval. There is of course no loss of generality involved in doing this, since the addition of a constant term is a trivial procedure that does not bear on the convergence issues. The discussion of the convergence of arbitrary trigonometric series $S$ is therefore equivalent to the separate discussion of the convergence of arbitrary cosine series $S_{\rm c}$ and arbitrary sine series $S_{\rm s}$. These last two classes consist of trigonometric series with definite parities and we will name them Definite-Parity trigonometric series, or DP trigonometric series for short.

A note about the fact that we limit ourselves to real Fourier series here. We might as well consider the series $S$ with complex coefficients $\alpha_{k}$ and $\beta_{k}$, but due to the linearity of the series with respect to these coefficients, any such complex Fourier series would at once decouple into two real Fourier series, one in the real part and one in the imaginary part, and would therefore reduce the discussion to the one we choose to develop here. Therefore nothing fundamentally new is introduced by the examination of complex Fourier series, and it is enough to limit the discussion to the real case.