Fourier Series

In [#!CAoRFI!#] we showed that, given any integrable real function $f(\theta)$, one can construct a corresponding inner analytic function $w(z)=u(\rho,\theta)+\mbox{\boldmath$\imath$}v(\rho,\theta)$, from the real part of which $f(\theta)$ can be recovered almost everywhere on the unit circle, through the use of the $\rho\to 1_{(-)}$ limit, where $(\rho,\theta)$ are polar coordinates on the complex plane. In that construction we started by calculating the Fourier coefficients [#!FSchurchill!#] of the real function, which is always possible given that the function is integrable, using the usual integrals defining these coefficients,

$$\alpha_{0} = \frac{1}{\pi} \int_{-\pi}^{\pi}d\theta\, f(\theta),$$

$$\alpha_{k} = \frac{1}{\pi} \int_{-\pi}^{\pi}d\theta\, \cos(k\theta)f(\theta),$$

$$\beta_{k} = \frac{1}{\pi} \int_{-\pi}^{\pi}d\theta\, \sin(k\theta)f(\theta),$$ (4)

for $k\in\{1,2,3,\ldots,\infty\}$. We then defined a set of complex Taylor coefficients

$$c_{0} = \frac{1}{2}\,\alpha_{0},$$

$$c_{k} = \alpha_{k} - \mbox{\boldmath$\imath$}\beta_{k},$$ (5)

for $k\in\{1,2,3,\ldots,\infty\}$. Next we defined a complex variable $z$ associated to $\theta$, using the positive real variable $\rho$, by $z=\rho\exp(\mbox{\boldmath$\imath$}\theta)$. Using all these elements we then constructed the power series

$$S(z) = \sum_{k=0}^{\infty} c_{k}z^{k},$$ (6)

which we showed to be convergent to an inner analytic function $w(z)=S(z)$ within the open unit disk. This power series is therefore the Taylor series of $w(z)$. We also proved that one recovers the real function $f(\theta)$ almost everywhere on the unit circle from the $\rho\to 1_{(-)}$ limit of the real part $u(\rho,\theta)$ of $w(z)$. It is now very easy to show that the Fourier series of an integrable real function $f(\theta)$ is simply given by the real part of this Taylor series, when restricted to the unit circle. Writing the series explicitly in terms of the polar coordinates $(\rho,\theta)$ of the complex plane, we get

$$w(z) = \frac{\alpha_{0}}{2} + \sum_{k=1}^{\infty} (\alpha_{k}-\mbox{\bol... ...k}) \rho^{k} \left[ \cos(k\theta)+\mbox{\boldmath$\imath$}\sin(k\theta) \right]$$

$$= \frac{\alpha_{0}}{2} + \sum_{k=1}^{\infty} \rho^{k} \left[ \alpha_{k}\cos(k\theta) + \beta_{k}\sin(k\theta) \right] +$$

$$\hspace{1.5em} + \mbox{\boldmath$\imath$} \sum_{k=1}^{\infty} \rho^{k} \left[ \alpha_{k}\sin(k\theta) - \beta_{k}\cos(k\theta) \right],$$ (7)

where $w(z)=u(\rho,\theta)+\mbox{\boldmath$\imath$}v(\rho,\theta)$. Taking now the $\rho\to 1_{(-)}$ limit we get

$$u(1,\theta) + \mbox{\boldmath$\imath$} v(1,\theta) = \frac{\alpha_{0}}{2} + \sum_{k=1}^{\infty} \left[ \alpha_{k}\cos(k\theta) + \beta_{k}\sin(k\theta) \right] +$$

$$\hspace{1.5em} + \mbox{\boldmath$\imath$} \sum_{k=1}^{\infty} \left[ \alpha_{k}\sin(k\theta) - \beta_{k}\cos(k\theta) \right].$$ (8)

It follows, therefore, that the real part of $w(z)$ for $\rho=1$ is the Fourier series of $f(\theta)$,

$$u(1,\theta) = \frac{\alpha_{0}}{2} + \sum_{k=1}^{\infty... ... \alpha_{k}\cos(k\theta) + \beta_{k}\sin(k\theta) \right],$$ (9)

and that the imaginary part of $w(z)$ for $\rho=1$ is the Fourier series of the real function which is the Fourier conjugate of $f(\theta)$,

$$v(1,\theta) = \sum_{k=1}^{\infty} \left[ \alpha_{k}\sin(k\theta) - \beta_{k}\cos(k\theta) \right].$$ (10)

Here we see that, with respect to the Fourier series of $u(1,\theta)$, the $k=0$ term is missing, all the other coefficients are the same, while the $\cos(k\theta)$ were exchanged for $\sin(k\theta)$, and the $\sin(k\theta)$ were exchanged for $-\cos(k\theta)$. In [#!CAoRFI!#] we proved that $u(1,\theta)$ is equal to $f(\theta)$ almost everywhere, irrespective of the convergence or lack of convergence of the Fourier series, so that it now becomes clear that, when and where this trigonometric series converges at all, it converges to the original integrable real function,

$$f(\theta) = \frac{\alpha_{0}}{2} + \sum_{k=1}^{\infty} ... ... \alpha_{k}\cos(k\theta) + \beta_{k}\sin(k\theta) \right].$$ (11)

The convergence of this Fourier series can be characterized in terms of the singularities of the inner analytic function $w(z)$ on the unit disk. If there are no singularities of $w(z)$ on the unit circle, then the maximum convergence disk of its Taylor series is larger than the unit disk, and contains it. Therefore, in this case the Fourier series is always convergent, as well as absolutely and uniformly convergent. On the other hand, if there is at least one singularity of $w(z)$ on the unit circle, then the unit disk is the maximum disk of convergence of the Taylor series, and in this case the Fourier series may or may not be convergent. In this case we see that, given any integrable real function, the issue of the convergence of its Fourier series is thus identified completely with the issue of the convergence of the Taylor series of the corresponding inner analytic function, at the border of its maximum convergence disk.

From the expansion in Equation (7) we see that the recovery of $f(\theta)$ from its Fourier coefficients via the inner analytic function $w(z)$, as we discussed in [#!CAoRFI!#], which works even when the Fourier series diverges almost everywhere, is equivalent to taking the $\rho\to 1_{(-)}$ limit of the following modified or regulated Fourier series,

$$f(\theta) = \lim_{\rho\to 1_{(-)}} \left\{ \frac{\alpha_{0}}{2} + \sum_{k=1}^... ...ho^{k} \left[ \alpha_{k}\cos(k\theta) + \beta_{k}\sin(k\theta) \right] \right\}$$

$$= \frac{\alpha_{0}}{2} + \lim_{\rho\to 1_{(-)}} \sum_{k=1}^{\infty} \rho^{k} \left[ \alpha_{k}\cos(k\theta) + \beta_{k}\sin(k\theta) \right],$$ (12)

which of course is always convergent, so long as $\rho<1$, for all integrable real functions $f(\theta)$, given that it is the real part of the convergent Taylor series of $w(z)$. The limit indicated will exist when and where $f(\theta)$ can be recovered from the real part of the corresponding inner analytic function. This holds for all the points on the unit circle where the inner analytic function $w(z)$ is either analytic or has only soft singularities. This recipe constitutes, therefore, a very general summation rule for Fourier series.