Extension of the Theory

Up to this point we have been examining only the Fourier theory of integrable real functions. In addition to this, a small extension of the theory has already been considered when we wrote the Fourier expansion of the Dirac delta ``function'' in Equations (27) and (28) of Section 4, with the help of the summation rule given in Equation (12). This ``function'' has in common with the integrable real functions the fact that its Fourier coefficients $\alpha_{k}$ and $\beta_{k}$ are limited when we take the limit $k\to\infty$. The same is true for the corresponding complex Taylor coefficients $c_{k}$ in either case. However, the correspondence between real Fourier coefficients and complex Taylor coefficients given by the relations in Equation (5) can be generalized, independently of any concerns about the behavior of these coefficients when $k\to\infty$, and independently of any concerns about the convergence of the corresponding series.

We will now discuss the extension of the Fourier theory beyond the realm of integrable real functions. One way to look at this, which is probably the most general possible way, is to simply consider the set of all inner analytic functions. Given any inner analytic function $w(z)$ and its complex Taylor series around the origin, which is therefore convergent within the open unit disk, and irrespective of whether or not $w(z)$ corresponds to an integrable real function, one can define a corresponding real Fourier series on the unit circle. In all such cases the issues of convergence of the resulting Fourier series are then completely identified with the corresponding issues for the Taylor series restricted to the unit circle, which is often the border of its maximum disk of convergence. Important examples which are not related to integrable real functions are the cases of the Dirac delta ``function'' and of its derivatives of all orders, which were discussed in detail in [#!CAoRFII!#].

Another way to look at this issue is through the properties of the sets of complex coefficients $c_{k}$ of the Taylor series. Given any set of complex coefficients $c_{k}$, regardless of whether or not they follow from a known inner analytic function, one can construct both a complex power series $S(z)$ and the corresponding real coefficients $\alpha_{k}$ and $\beta_{k}$, using the relations in Equations (5) and (6). In many cases the Fourier series generated by these real coefficients will not converge, even if the complex power series converges to an inner analytic function within the open unit disk. However, if the complex power series is indeed convergent on that disk, then one can discuss whether or not a real object can be defined on the unit circle, through the $\rho\to 1_{(-)}$ limit from the open unit disk, for example using the summation rule for Fourier series given in Equation (12).

If we examine that summation rule, it is apparent that it will work for much more than just integrable real functions, which always have bounded Fourier coefficients. For example, one may have unbounded Fourier coefficients $\alpha_{k}$ and $\beta_{k}$, such as those of the $n^{\rm th}$ derivative of the delta ``function'', which diverge to infinity as the power $k^{n}$ when $k\to\infty$, and still have a well-defined inner analytic function, as was shown in detail in [#!CAoRFII!#]. In fact, one can show that the summation rule can be used for all sets of Fourier coefficients that do not diverge exponentially fast with $k$. In order to develop this idea, let us first define a very general condition on the sequences of complex coefficients that guarantees that the corresponding power series are convergent within the open unit disk, and thus converge to inner analytic functions.

Definition 1   : Exponentially Bounded Coefficients

Given an arbitrary ordered set of complex coefficients $a_{k}$, for $k\in\{0,1,2,3,\ldots,\infty\}$, if they satisfy the condition that

$$\lim_{k\to\infty} \vert a_{k}\vert\,{\rm e}^{-Ck} = 0,$$ (45)

for all real $C>0$, then we say that the sequence of coefficients $a_{k}$ is exponentially bounded.

What this means is that $a_{k}$ may or may not go to zero as $k\to\infty$, may approach a non-zero complex number, and may even diverge to infinity as $k\to\infty$, so long as it does not do so exponentially fast. This includes therefore not only the sequences of complex Taylor coefficients corresponding to all possible convergent Fourier series, but many sequences that correspond to Fourier series that diverge almost everywhere. Also, it not only includes the sequences of complex Taylor coefficients corresponding to all possible integrable real functions, but many sequences of coefficients that cannot be obtained at all from a real function, such as those associated to the Dirac delta ``function'' and its derivatives of arbitrarily high orders, as was shown in [#!CAoRFII!#]. We see therefore that this is a very weak condition on the complex sequence of coefficients $a_{k}$.

Before we proceed to the extension of the Fourier theory, let us establish a preliminary result, which can be understood as a property of the sequences of complex coefficients $a_{k}$ which satisfy the condition stated in Definition 1. We will show that the condition expressed in Equation (45) implies an infinite collection of other similar conditions involving the $k\to\infty$ limit, that express modified bounds on these sequences of coefficients.

Property 1.1   : If the sequence of complex coefficients $a_{k}$ is exponentially bounded, then we also have that

$$\lim_{k\to\infty} \vert a_{k}\vert k^{p}\,{\rm e}^{-Ck} = 0,$$ (46)

for all real $C>0$ and for all real powers $p>0$.

This is just a formalization of the well-known fact that the negative-exponent real exponential function of $k$ goes to zero faster than any positive real power of $k$ goes to infinity, as $k\to\infty$. In order to prove this, we observe that for $k>0$ we may write the function of $k$ on the left-hand side of Equation (46) as

$$\vert a_{k}\vert k^{p}\,{\rm e}^{-Ck} = \vert a_{k}\vert\,{\rm e}^{\,p\ln(k)}\,{\rm e}^{-Ck}.$$ (47)

Note that this is a positive real quantity. Recalling the properties of the real logarithm function, we now observe that, given an arbitrary real number $A>0$, there is always a sufficiently large finite value $k_{m}$ of $k$ above which $\ln(k)<Ak$. Due to this we may write, for all $k>k_{m}$,

$$\vert a_{k}\vert k^{p}\,{\rm e}^{-Ck} < \vert a_{k}\vert\,{\rm e}^{\,pAk}\,{\rm e}^{-Ck},$$ (48)

since the exponential with a strictly positive real exponent is a monotonically increasing function. If we now choose $A=C/(2p)$, which we may do because this value is positive and not zero, we get that, for all $k>k_{m}$,

$$\vert a_{k}\vert k^{p}\,{\rm e}^{-Ck} < \vert a_{k}\vert\,{\rm e}^{Ck/2}\,{\rm e}^{-Ck}$$

$$= \vert a_{k}\vert\,{\rm e}^{-Ck/2}.$$ (49)

According to our hypothesis about the coefficients $a_{k}$, the $k\to\infty$ limit of the expression in the right-hand side is zero for any strictly positive value of $C'=C/2$, so that taking the $k\to\infty$ limit we establish our preliminary result,

$$\lim_{k\to\infty} \vert a_{k}\vert k^{p}\,{\rm e}^{-Ck} = 0,$$ (50)

for all real $C>0$ and all real $p>0$. Therefore, we have established this property.

Let us now show that the condition that the sequence of complex coefficients $c_{k}$ in Equation (5) is exponentially bounded is equivalent to the condition that the sequences of real coefficients $\alpha_{k}$ and $\beta_{k}$ are both exponentially bounded. First, if we assume that the sequences $\alpha_{k}$ and $\beta_{k}$ are both exponentially bounded, and since from Equation (5) we have that

$$\vert c_{k}\vert = \sqrt{\vert\alpha_{k}\vert^{2}+\vert\beta_{k}\vert^{2}},$$ (51)

it follows at once that

$$\lim_{k\to\infty} \vert c_{k}\vert\,{\rm e}^{-Ck} = \lim_{k\to\infty} \sqrt{ \left(\vert\alpha_{k}\vert\,{\rm e}^{-Ck}\right)^{2} + \left(\vert\beta_{k}\vert\,{\rm e}^{-Ck}\right)^{2} }$$

$$= \sqrt{ \left(\lim_{k\to\infty}\vert\alpha_{k}\vert\,{\rm e}^{-Ck}... ...t)^{2} + \left(\lim_{k\to\infty}\vert\beta_{k}\vert\,{\rm e}^{-Ck}\right)^{2} }$$

$$= 0,$$ (52)

since both limits in the right-hand side are zero, thus establishing that the sequence $c_{k}$ is exponentially bounded. Second, if we assume that the sequence $c_{k}$ is exponentially bounded, and since from Equation (5) we have that

$$\vert c_{k}\vert = \sqrt{\vert\alpha_{k}\vert^{2}+\vert\beta_{k}\vert^{2}}$$

$$\geq \vert\alpha_{k}\vert \;\;\;\Rightarrow$$

$$\vert c_{k}\vert\,{\rm e}^{-Ck} \geq \vert\alpha_{k}\vert\,{\rm e}^{-Ck},$$ (53)

taking the $k\to\infty$ limit and using the assumed property of the sequence of coefficients $c_{k}$ it follows that

$$\lim_{k\to\infty} \vert c_{k}\vert\,{\rm e}^{-Ck} \geq \lim_{k\to\infty} \vert\alpha_{k}\vert\,{\rm e}^{-Ck} \;\;\;\Rightarrow$$

$$0 \geq \lim_{k\to\infty} \vert\alpha_{k}\vert\,{\rm e}^{-Ck} \;\;\;\Rightarrow$$

$$\lim_{k\to\infty} \vert\alpha_{k}\vert\,{\rm e}^{-Ck} = 0,$$ (54)

thus establishing that the sequence $\alpha_{k}$ is exponentially bounded. Clearly, an identical argument can be made for the sequence $\beta_{k}$. This establishes that the statement that the sequence of complex coefficients $c_{k}$ is exponentially bounded is equivalent to the statement that the sequences of real coefficients $\alpha_{k}$ and $\beta_{k}$ are both exponentially bounded.

Let us now prove the following theorem about the convergence of the power series constructed out of a given arbitrary sequence of complex coefficients $c_{k}$.

Theorem 2   : If the sequence of complex coefficients $c_{k}$, for $k\in\{0,1,2,3,\ldots,\infty\}$, is exponentially bounded, then the power series constructed from this sequence of coefficients converges within the open unit disk.

Given the arbitrary sequence of complex coefficients $c_{k}$, we may construct the complex power series in the complex $z$ plane, just as we did in [#!CAoRFI!#],

$$S(z) = \sum_{k=0}^{\infty} c_{k}z^{k}.$$ (55)

We will first show that, if the sequence of coefficients $c_{k}$ is exponentially bounded, then this series is absolutely convergent inside the open unit disk, which then implies that it is simply convergent there.

Proof 2.1   :

In order to prove that $S(z)$ is absolutely convergent, we consider the real power series $\overline{S}(z)$ of the absolute values of the terms of that series, which we write as

$$\overline{S}(z) = \sum_{k=0}^{\infty} \vert c_{k}\vert\rho^{k}$$

$$= \sum_{k=0}^{\infty} \vert c_{k}\vert\,{\rm e}^{k\ln(\rho)}.$$ (56)

Since $\rho<1$ inside the open unit disk, the logarithm shown is strictly negative, and we may put $\ln(\rho)=-C$ with real $C>0$. We can now see that, according to our hypothesis about the coefficients $c_{k}$, the terms of this series go to zero as $k\to\infty$,

$$\overline{S}(z) = \sum_{k=0}^{\infty} \vert c_{k}\vert\,{\rm e}^{-Ck},$$ (57)

since $C$ is real and strictly positive. In order to establish the convergence of this real series, we write

$$\overline{S}(z) = \vert c_{0}\vert + \sum_{k=1}^{\infty} \frac{k^{2}\vert c_{k}\vert\,{\rm e}^{-Ck}}{k^{2}}.$$ (58)

According to the property expressed in Equation (46), with $p=2$, the numerator shown above goes to zero as $k\to\infty$, and therefore above a sufficiently large value $k_{m}$ of $k$ it is less that one, so that we may write that

$$\overline{S}(z) = \sum_{k=0}^{k_{m}} \vert c_{k}\vert\,{\rm e}^{-Ck} + \sum_{k=k_{m}+1}^{\infty} \frac{k^{2}\vert c_{k}\vert\,{\rm e}^{-Ck}}{k^{2}}$$

$$< \sum_{k=0}^{k_{m}} \vert c_{k}\vert\,{\rm e}^{-Ck} + \sum_{k=k_{m}+1}^{\infty} \frac{1}{k^{2}}.$$ (59)

The first term on the right-hand side is a finite sum and therefore is finite, and the second term can be bounded from above by a convergent asymptotic integral on $k$, so that we have

$$\overline{S}(z) < \sum_{k=0}^{k_{m}} \vert c_{k}\vert\,{\rm e}^{-Ck} + \int_{k_{m}}^{\infty}dk\, \frac{1}{k^{2}}$$

$$= \sum_{k=0}^{k_{m}} \vert c_{k}\vert\,{\rm e}^{-Ck} + \frac{-1}{k}\left.\rule{0em}{3ex}\right[_{\,k_{m}}^{\,\infty}$$

$$= \sum_{k=0}^{k_{m}} \vert c_{k}\vert\,{\rm e}^{-Ck} + \frac{1}{k_{m}}.$$ (60)

This last expression is therefore a finite upper bound for all the partial sums of the series $\overline{S}(z)$. It follows that $\overline{S}(z)$, which is a real sum of positive terms, so that its partial sums form a monotonically increasing real sequence which is now found to be bounded from above, is therefore convergent. It then follows that $S(z)$ is absolutely convergent and therefore convergent. Since this is valid for all $\rho<1$, we may conclude that $S(z)$ converges on the open unit disk. This completes the proof of Theorem 2.

Since the series $S(z)$ considered above is a convergent power series within the open unit disk, it converges to an analytic function $w(z)$ in that domain, which is therefore an inner analytic function. We therefore conclude that, if the sequence of complex coefficients $c_{k}$ in Equation (5) is exponentially bounded, then it is the set of Taylor coefficients of an inner analytic function. It now follows that, if the corresponding Fourier coefficients $\alpha_{k}$ and $\beta_{k}$ are both exponentially bounded, then the corresponding complex coefficients $c_{k}$ are also exponentially bounded, and therefore the corresponding Fourier series can be regulated by the use of the summation rule in Equation (12). Unless the Fourier coefficients go to zero as $k\to\infty$, the Fourier series on the unit circle is sure to diverge almost everywhere. One can then consider defining the corresponding real object on the unit circle using the $\rho\to 1_{(-)}$ limit from the open unit disk, for example through the use of the summation rule for the Fourier series, given in Equation (12).

In this way the Fourier theory of integrable real functions on the unit circle can be extended to a much larger set of real objects, including for example all the singular distributions discussed in [#!CAoRFII!#], as well as the examples of non-integrable real functions mentioned in that paper. In fact, this extension of the Fourier theory includes a large class of non-integrable real functions, as will be shown in the fourth paper of this series. In this extended Fourier theory the real objects can be considered as representable directly by their sequences of Fourier coefficients, even when the corresponding Fourier series diverge. All operations involving these divergent Fourier series can be mapped to absolutely and uniformly convergent series and analytic operations within the open unit disk, whose results are then taken to the unit circle through the use of the $\rho\to 1_{(-)}$ limit. In many simple cases the mere values of the real objects on the unit circle will be recovered in this way, and in other more abstract cases global properties of the real objects may be obtained in this way, such as in the case of the Dirac delta ``function'' and its derivatives of all orders, as was discussed in detail in [#!CAoRFII!#].