Notes on the Convergence Problem

In this paper we have made the deliberate choice of not discussing the question of the convergence of Fourier series in any amount of detail, that is, we have not discussed any of the many existing so-called Fourier theorems. The reason for this is that we believe that this would constitute a rather long and complex discussion, best left for a separate paper. Instead, we have focused our attention on the summation rule given in Equation (12), according to which all Fourier series of integrable real functions, without any further restrictions, can be added up in such a way that one is able to recover the functions from their Fourier coefficients, even if the Fourier series themselves diverge. However, we may make a few comments about the issue of convergence, without going too far afield in that subject, in order to exhibit the relation between our complex analytic structure and the convergence problem.

First of all, let us recall that, as was shown in [#!CAoRFI!#], the real function $f(\theta)$ is equal almost everywhere to the real part of the corresponding inner analytic function $w(z)$, taken in the $\rho\to 1_{(-)}$ limit, and also that, as we have shown in Section 2 of this paper, the Fourier series of $f(\theta)$ is given by the real part of the Taylor series $S(z)$ of $w(z)$ in that same limit. Therefore, it is clearly apparent that, as was already noted in Section 2, the problem of the convergence of real Fourier series is completely identified with the problem of the convergence of the corresponding complex power series on the unit circle, including the cases in which it is the rim of their maximum disks of convergence. Whatever is established for one type of series is also valid for the other. As was also noted in Section 2, the convergence properties on the unit circle will depend on the existence and nature of the singularities of $w(z)$ on that circle.

One way to discuss the issue of convergence is to observe that the summation rule given in Equation (12) involves two limits, one being the series summation limit and the other being the $\rho\to 1_{(-)}$ limit from the interior of the unit disk to the unit circle. What has been shown so far in this series of papers is that if one takes the series summation limit first, and only after that the $\rho\to 1_{(-)}$ limit, then it is always possible to recover the real function from its Fourier coefficients. It is therefore immediately apparent that the statement that the Fourier series converges over the unit circle is equivalent to the statement that the order of these two limits can be inverted. In fact, by first taking the $\rho\to 1_{(-)}$ limit one obtains the usual Fourier series over the unit circle, and if one is then able to take the series summation limit, then that series converges to the corresponding real function.

The general problem of deciding under what conditions the order of the two limits can be inverted is not a simple one. However, it is not too difficult to use our analytic structure to write the partial sums of the Fourier series in terms of real integrals which are similar to the Dirichlet integrals usually involved in some of the Fourier theorems. This can then be used as the starting point for further discussions of the convergence problem, including in particular discussions establishing the connection of the analytic structure with specific Fourier theorems. In order to do this, let $f(\theta)$ be an integrable real function on $[-\pi,\pi]$ and let the real numbers $\alpha_{0}$, $\alpha_{k}$ and $\beta_{k}$, for $k\in\{1,2,3,\ldots,\infty\}$, be its Fourier coefficients. We may define the complex coefficients $c_{0}$ and $c_{k}$ shown in Equation (5), and thus construct the corresponding inner analytic function $w(z)$ within the open unit disk, using the power series $S(z)$ given in Equation (6), which, as was shown in [#!CAoRFI!#], always converges for $\vert z\vert<1$. The partial sums of the first $N$ terms of this series are given by

$$S_{N}(z) = \sum_{k=0}^{N-1} c_{k}z^{k},$$ (34)

a complex sequence which, for $\vert z\vert<1$, we already know to converge to $w(z)$ in the $N\to\infty$ limit. Note however that, since $S_{N}(z)$ is in fact an analytic function over the whole complex plane, this expression itself can be consistently considered for all $z$, and in particular for $z$ on the unit circle, where $\vert z\vert=1$. Note also that the function $w(z)$ may have singularities on the unit circle, but that these must be integrable ones, at least along that circle. In addition to this, the complex coefficients $c_{k}$ may be written as integrals involving $w(z)$, with the use of the Cauchy integral formulas,

$$c_{k} = \frac{1}{2\pi\mbox{\boldmath$\imath$}} \oint_{C}dz\, \frac{w(z)}{z^{k+1}},$$ (35)

where $C$ can be taken as a circle centered at the origin, with radius $\rho\leq 1$. The reason why we may include the case $\rho=1$ here is that, as was shown in [#!CAoRFI!#], as a function of $\rho$ the expression above for $c_{k}$ is not only constant within the open unit disk, but also continuous from within at the unit circle. In this way the coefficients $c_{k}$ may be written back in terms of the inner analytic function $w(z)$. If we substitute this expression for $c_{k}$ back in the partial sums of the series we get

$$S_{N}(z) = \sum_{k=0}^{N-1} z^{k}\, \frac{1}{2\pi\mbox{\boldmath$\imath$}} \oint_{C}dz_{1}\, \frac{w(z_{1})}{z_{1}^{k+1}}$$

$$= \frac{1}{2\pi\mbox{\boldmath$\imath$}} \oint_{C}dz_{1}\, \frac{w(z_{1})}{z_{1}} \sum_{k=0}^{N-1} \left( \frac{z}{z_{1}} \right)^{k},$$ (36)

where we must have $\vert z_{1}\vert\leq 1$. The sum is now a finite geometric progression, so that we have

$$S_{N}(z) = \frac{1}{2\pi\mbox{\boldmath$\imath$}} \oint_{C}dz_{1}\, \frac{w(z_{1})}{z_{1}}\, \frac {1-(z/z_{1})^{N}} {1-(z/z_{1})}$$

$$= \frac{1}{2\pi\mbox{\boldmath$\imath$}} \oint_{C}dz_{1}\, \frac{w(... ...mbox{\boldmath$\imath$}} \oint_{C}dz_{1}\, \frac{w(z_{1})}{z_{1}^{N}(z_{1}-z)}.$$ (37)

A careful discussion of this formula is now in order. There are two relevant cases to consider. In the first case we see that, if we have that $\vert z_{1}\vert>\vert z\vert$, then in the first term above we obtain the expression of the Cauchy integral formula for $w(z)$, which then allows us to write an explicit expression for the remainder of the complex power series after one adds up its first $N$ terms,

$$R_{N}(z) = w(z) - S_{N}(z)$$

$$= \frac{z^{N}}{2\pi\mbox{\boldmath$\imath$}} \oint_{C}dz_{1}\, \frac{w(z_{1})}{z_{1}^{N}(z_{1}-z)},$$ (38)

where $\vert z\vert<\vert z_{1}\vert\leq 1$. This expression of the remainder in closed form, an expression which, as one can easily show, goes to zero in the $N\to\infty$ limit, is what makes it easy to discuss the convergence of complex power series. However, this expression does not give us an equivalent expression for the remainder of the Fourier series, because this would require us to make $\vert z\vert=\vert z_{1}\vert=1$, which is not allowed by the strict inequality $\vert z\vert<\vert z_{1}\vert$, a restriction which is due to the use of the Cauchy integral formulas. In the second case we observe that, if we have that $\vert z_{1}\vert<\vert z\vert$, then the first term in Equation (37) is simply zero, and therefore we get a modified expression for the partial sums of the series,

$$S_{N}(z) = -\, \frac{z^{N}}{2\pi\mbox{\boldmath$\imath$}} \oint_{C}dz_{1}\, \frac{w(z_{1})}{z_{1}^{N}(z_{1}-z)},$$ (39)

where $\vert z\vert>\vert z_{1}\vert$. Note that in this case we are unable to write an explicit expression in closed form for the remainder of the series, a fact which seems to be related to the remarkable difficulty in finding a necessary and sufficient condition for the convergence of Fourier series. Since $z$ may have any complex value in this expression, we may now make $z=\rho\exp(\mbox{\boldmath$\imath$}\theta)$ with $\rho=1$, as well as $z_{1}=\rho_{1}\exp(\mbox{\boldmath$\imath$}\theta_{1})$, and thus write the integral explicitly in terms of the variable $\theta_{1}$ on the circle of radius $\rho_{1}$,

$$S_{N}(1,\theta) = -\, \frac{\,{\rm e}^{\mbox{\boldmath\scriptsize$\imath$}N\theta}}... ...th$}\theta_{1}}-\,{\rm e}^{\mbox{\boldmath\scriptsize$\imath$}\theta} \right) }$$

$$= -\, \frac{1}{2\pi\rho_{1}^{N-1}} \int_{-\pi}^{\pi}d\theta_{1}\, \... ... \rho_{1}-\,{\rm e}^{\mbox{\boldmath\scriptsize$\imath$}(\theta-\theta_{1})} }.$$ (40)

Making $\Delta\theta=\theta_{1}-\theta$ we have

$$S_{N}(1,\theta) = -\, \frac{1}{2\pi\rho_{1}^{N-1}} \int... ...{\rm e}^{-\mbox{\boldmath\scriptsize$\imath$}\Delta\theta} }.$$ (41)

In order to be able to write explicitly the real and imaginary parts of the partial sums, we must now rationalize this expression,

$$S_{N}(1,\theta) = -\, \frac{1}{2\pi\rho_{1}^{N-1}} \int_{-\pi}^{\pi}d\theta_{1}\, \... ... \rho_{1}-\,{\rm e}^{\mbox{\boldmath\scriptsize$\imath$}\Delta\theta} \right) }$$

$$= \frac{1}{2\pi\rho_{1}^{N-1}} \int_{-\pi}^{\pi}d\theta_{1}\, w(\rh... ...})\, \,{\rm e}^{-\mbox{\boldmath\scriptsize$\imath$}(N-1/2)\Delta\theta} \times$$

$$\hspace{7em} \times\, \frac { \,{\rm e}^{\mbox{\boldmath\scriptsi... ...tsize$\imath$}\Delta\theta/2} } { 1+\rho_{1}^{2}-2\rho_{1}\cos(\Delta\theta) }.$$ (42)

The expression can be somewhat simplified if we write most things in terms of $\Delta\theta/2$, as well as in terms of $N_{1}=N-1/2$,

$$S_{N}(1,\theta) = \frac{1}{2\pi\rho_{1}^{N-1}} \int_{-\pi}^{\pi}d\theta_{1}\, \left... ...1},\theta_{1}) + \mbox{\boldmath$\imath$} v(\rho_{1},\theta_{1}) \right] \times$$

$$\hspace{7em} \times \left[ \rule{0em}{2.5ex} \cos(N_{1}\Delta\theta) - \mbox{\boldmath$\imath$} \sin(N_{1}\Delta\theta) \right] \times$$ (43)

$$\hspace{7em} \times\, \frac { (1-\rho_{1})\cos(\Delta\theta/2) + ... ...)\sin(\Delta\theta/2) } { (1-\rho_{1})^{2}+4\rho_{1}\sin^{2}(\Delta\theta/2) }.$$

In this context, a Fourier theorem is one which states sufficient conditions on $f(\theta)$ under which it follows that the real part of the corresponding sequence of partial sums $S_{N}(1,\theta)$ converges in the $N\to\infty$ limit, after one takes the $\rho_{1}\to 1$ limit, so that the integral is written over the unit circle. In any circumstances in which one managed to calculate these integrals explicitly in terms of $\rho_{1}$, for $\rho_{1}<1$, one would then be able to consider taking the $\rho_{1}\to 1$ limit of the resulting expression. However, despite the facts that $f(\theta)=u(1,\theta_{1})$ and that $g(\theta)=v(1,\theta_{1})$, almost everywhere over the unit circle, as well as the fact that these are integrable real functions, we cannot simply take the $\rho_{1}\to 1$ limit of this expression as it stands, because it was derived under the hypothesis that $\vert z\vert>\vert z_{1}\vert$, and therefore that $\rho>\rho_{1}$, which at this point implies the strict inequality $1>\rho_{1}$. We may however put $\rho_{1}=1$ in the integrand simply in order to simplify the integrals, so as to exhibit their structure more clearly. If one does that one obtains

$$\int_{-\pi}^{\pi}d\theta_{1}\, \left[ \rule{0em}{2.5ex} ... ...2ex}(N-1/2)\Delta\theta\right] } { \sin(\Delta\theta/2) },$$ (44)

which clearly reduces to Dirichlet integrals and other similar integrals. A more complete discussion of the issue of convergence would require considerable development of the ideas and structures involved in these arguments. It is currently not entirely clear how useful the analytic structure within the open unit disk can be in regards to proving known Fourier theorems or discovering new ones.