Representation of Non-Integrable Real Functions

First of all, let us describe, in general lines, the algorithm we propose to use in order to determine an inner analytic function that corresponds to a given non-integrable real function. Given a non-integrable real function $f(\theta)$, which is however locally integrable almost everywhere, and whose hard singularities have a finite maximum degree of hardness $n$, we sectionally integrate it $n$ times. Since by hypothesis all the non-integrable singularities of $f(\theta)$ have degrees of hardness of at most $n$, the resulting function $f^{-n\prime}(\theta)$ is in fact an integrable one on the whole unit circle. Therefore, we may use it to construct the corresponding inner analytic function, which we will name $w^{-n\mbox{\Large$\cdot$}\!}(z)$, using the construction presented in [#!CAoRFI!#]. Having this inner analytic function, we then calculate its $n^{\rm th}$ angular derivative, in order to obtain $w(z)$, which is the inner analytic function that corresponds to the non-integrable real function $f(\theta)$. However, the $n$-fold angular differentiation process produces only the proper inner analytic function $w_{p}(z)$ associated to $w(z)$, and therefore produces $f(\theta)$ only up to an overall constant related to the whole unit circle, as we will soon see. Of course this scheme will succeed if and only if the non-integrable singular points of $f(\theta)$ all have finite degrees of hardness, with a maximum value of $n$.

In this section we will prove the following theorem.

Theorem 1   : Every non-integrable real function defined almost everywhere on the periodic interval, which is locally integrable almost everywhere, and which is such that its non-integrable singularities have finite degrees of hardness, can be represented by an inner analytic function, and can be recovered almost everywhere on its domain of definition by means of the limit to the unit circle of the real part of that inner analytic function.

Before we attempt to prove the theorem, let us establish some notation for a sectionally defined real function $f(\theta)$, that will be similar to the one adopted for the piecewise polynomial functions discussed in [#!CAoRFII!#], which is also given in Equation (6). Since a non-integrable real function $f(\theta)$ which is locally integrable almost everywhere is not defined at the singular points corresponding to the angles $\theta_{i}$, for $i\in\{1,\ldots,N\}$, it is in fact only sectionally defined, in $N$ open intervals between consecutive singularities, as given in Equation (5). Let us therefore specify the definition of $f(\theta)$ as a set of $N\geq 1$ functions $f_{i}(\theta)$ defined on the $N$ sections $(\theta_{i},\theta_{i+1})$,

$$f(\theta) = \left\{ \rule{0em}{2.5ex} f_{i}(\theta),i\in\{1,\ldots,N\} \right\},$$ (11)

where, as before, we adopt the convention that every section $(\theta_{i},\theta_{i+1})$ is numbered after the singular point $\theta_{i}$ at its left end.

As a preliminary to the proof of the theorem, some considerations are in order, regarding the multiple sectional integration of such non-integrable functions. Note that, if $f(\theta)$ is locally integrable almost everywhere, then it is integrable within each one of the $N$ open intervals that define the sections. More precisely, it is integrable on every closed interval contained within one of these open intervals. In terms of the sectional functions, since $f_{i}(\theta)$ is integrable within its section, we may define a piecewise primitive for $f(\theta)$, by simply integrating each function $f_{i}(\theta)$ within the corresponding open interval, starting at some arbitrary reference point $\theta_{0,i}$ strictly within that open interval. This will define the piecewise primitive

$$f^{-1\prime}(\theta) = \left\{ \rule{0em}{2.5ex} f_{i}^{-1\prime}(\theta),i\in\{1,\ldots,N\} \right\},$$

$$f_{i}^{-1\prime}(\theta) = \int_{\theta_{0,i}}^{\theta}d\theta'\, f_{i}(\theta'),$$ (12)

where we have that $\theta_{i}<\theta<\theta_{i+1}$ and that $\theta_{i}<\theta_{0,i}<\theta_{i+1}$. Note that, since $f_{i}(\theta)$ is integrable on every closed interval contained within its section, it follows that $f_{i}^{-1\prime}(\theta)$ is limited on these closed intervals, and therefore is also integrable on them. Therefore, this process of sectional integration of the sectional functions can be iterated indefinitely. If we iterate this process of sectional integration, we obtain further piecewise primitives $f^{-2\prime}(\theta)$, $f^{-3\prime}(\theta)$, and so on. Note also that, during this process of multiple sectional integration, some non-integrable singularities, having a lower degree of hardness, may become integrable before the others. In this case we might ignore the singularities which became integrable, from that point on in the multiple integration process, which therefore effectively reduces the number of sections, but for simplicity we choose not to do that, and thus to keep the set of sections constant. Note that, in any case, we do continue the process of sectional integration until all the singularities have become integrable, of course.

Although for definiteness we are integrating from some particular reference points $\theta_{0,i}$ within each section, our objective here is actually to construct primitives, and therefore we may ignore the particular values chosen for $\theta_{0,i}$ if at each step in this iterative process we add an arbitrary integration constant to the primitive in each section, so that after $n$ such successive integrations a polynomial of degree $n-1$, with $n$ arbitrary coefficients, will have been added to the $n^{\rm th}$ primitive in the $i^{\rm th}$ section. We will express this as follows,

$$f^{-n\prime}(\theta) = \left\{ \rule{0em}{2.5ex} f_{i}^{-n\prime}(\theta) + P_{i}^{(n-1)}(\theta), i\in\{1,\ldots,N\} \right\}$$

$$= \left\{ \rule{0em}{2.5ex} f_{i}^{-n\prime}(\theta), i\in\{1,\ldots,N\} \right\} + P_{(n-1)}(\theta),$$

$$P_{(n-1)}(\theta) = \left\{ \rule{0em}{2.5ex} P_{i}^{(n-1)}(\theta), i\in\{1,\ldots,N\} \right\},$$ (13)

where $f^{-n\prime}(\theta)$ is the most general piecewise $n^{\rm th}$ primitive of $f(\theta)$, $f_{i}^{-n\prime}(\theta)$ is an arbitrary $n^{\rm th}$ primitive of $f_{i}(\theta)$ in the $i^{\rm th}$ section, $P_{i}^{(n-1)}(\theta)$ is an arbitrary polynomial of order $n-1$ in the $i^{\rm th}$ section, and $P_{(n-1)}(\theta)$ is a piecewise polynomial function of order $n-1$, containing therefore $n$ arbitrary constants in each section. Note that $P_{(n-1)}(\theta)$ is always an integrable real function. Note also that, upon subsequent $n$-fold differentiation of $f^{-n\prime}(\theta)$ with respect to $\theta$, all the arbitrary constants that were added during the multiple integration process are then eliminated, the arbitrary polynomials vanish from the result, and we thus get back the original function $f(\theta)$, within the open intervals that constitute the sections,

Finally, we must emphasize some facts about the behavior of the correspondence between inner analytic functions an integrable real functions under the respective operations of differentiation and integration, that take us along the corresponding integral-differential chain. First, let us recall that, as we saw in [#!CAoRFI!#], both angular differentiation and angular integration produce only proper inner analytic functions, and thus always result in null Taylor coefficients $c_{0}$, and thus in null Fourier coefficients $\alpha_{0}$. Therefore, when we use angular integration and differentiation in our algorithm, we lose all information about these two $k=0$ coefficients. Second, as we also saw in [#!CAoRFI!#], the integration on $\theta$ implies that the resulting Fourier coefficient $\alpha_{0}$ becomes indeterminate due to the presence of an arbitrary integration constant. It is due to this that we must add arbitrary constants during the process of multiple sectional integration, thus generating the arbitrary piecewise polynomial real function $P_{(n-1)}(\theta)$.

At this point, let us review the algorithm we are to use here. First, starting from the non-integrable real function $f(\theta)$ we go $n$ steps along the integration direction of the integral-differential chain, using sectional integration on $\theta$ on the unit circle. This produces the $n$-fold piecewise primitive $f^{-n\prime}(\theta)$ containing the arbitrary piecewise polynomial real function $P_{(n-1)}(\theta)$. From the globally integrable real function $f^{-n\prime}(\theta)$ we then construct the inner analytic function $w^{-n\mbox{\Large$\cdot$}\!}(z)$, using the construction presented in [#!CAoRFI!#]. We then come back in the differentiation direction of the integral-differential chain the same number of steps, using this time angular differentiation of the inner analytic functions within the open unit disk. This produces an inner analytic function associated to $f(\theta)$, except for its coefficient $c_{0}$, which means that we recover only a proper inner analytic function, given that it is the result of a series of angular differentiations, and we will therefore denote this function by $w_{p}(z)=u_{p}(\rho,\theta)+\mbox{\boldmath$\imath$} v_{p}(\rho,\theta)$. Since this is equivalent to differentiation with respect to $\theta$ of the piecewise polynomial real function $P_{(n-1)}(\theta)$, it completely eliminates this function within the open intervals that constitute the sections. However, it also produces the superposition of a set of delta ``functions'' and derivatives of delta ``functions'' with support on the singular points between successive sections.

Thus we must conclude that this algorithm necessarily results in an indeterminate Taylor coefficient $c_{0}$ of the inner analytic function $w(z)$ which corresponds to the real function $f(\theta)$, so that we recover only the corresponding proper inner analytic function $w_{p}(z)$, and therefore it results in an equally indeterminate Fourier coefficients $\alpha_{0}$ related to the real function $f(\theta)$. Se see, therefore, that in this paper we are compelled to relax, to some extent, and only during the process of construction of the inner analytic functions, the correspondence between the real functions and these inner analytic functions, considering only proper inner analytic functions, and accepting the fact that they will correspond to the non-integrable real functions only up to a global constant over the whole unit circle. Once the proper inner analytic function $w_{p}(z)$ corresponding to a non-integrable real function $f(\theta)$ has been determined, the relation will be written as

$$f(\theta) \longleftrightarrow \frac{\alpha_{0}}{2} + w_{p}(z),$$ (14)

where $\alpha_{0}$ is a real constant, which can be determined afterwards, by the simple comparison of the known value of the real part $u_{p}(1,\theta)$ of $w_{p}(z)$ and the known value of $f(\theta)$, at any point $\theta$ on the unit circle where they do not have hard singularities. If $\theta_{0}$ is such a point, then we have that $\alpha_{0}=2[f(\theta_{0})-u_{p}(1,\theta_{0})]$. Once the coefficient $\alpha_{0}$ is thus determined, this finally determines completely the inner analytic function $w(z)$ that corresponds to $f(\theta)$,

$$w(z) = \frac{\alpha_{0}}{2} + w_{p}(z).$$ (15)

We are now ready to prove the theorem.

Proof 1.1   :

In order to prove the theorem, our first task is to show that we can obtain an inner analytic function $w(z)$ from the non-integrable real function $f(\theta)$, which is assumed to be locally integrable almost everywhere. Consider that we execute the iterative piecewise integration process described before $n$ times on $f(\theta)$, where $n$ is the maximum among all the degrees of hardness of the non-integrable hard singularities involved. By doing this we generate a real function $f^{-n\prime}(\theta)$ which has only soft or at most borderline hard singularities on the unit circle, and which is therefore integrable on the whole unit circle.

It follows therefore that we may determine its set of Fourier coefficients, as was done in [#!CAoRFI!#], which we will name $\alpha_{0}^{(-n)}$, $\alpha_{k}^{(-n)}$ and $\beta_{k}^{(-n)}$, for $k\in\{1,2,3,\ldots,\infty\}$. From this set of Fourier coefficients we may then define, again as was done in [#!CAoRFI!#], the complex Taylor coefficients $c_{k}^{(-n)}$, for $k\in\{0,1,2,3,\ldots,\infty\}$. From these coefficients we may then determine the unique inner analytic function that corresponds to the $n^{\rm th}$ piecewise primitive $f^{-n\prime}(\theta)$, which we will name $w^{-n\mbox{\Large$\cdot$}\!}(z)=u^{-n\prime}(\rho,\theta)+\mbox{\boldmath$\imath$}v^{-n\prime}(\rho,\theta)$. Note that we may use this notation unequivocally because every proper inner analytic function belongs to an infinite integral-differential chain, extending indefinitely to either side, so that we know that there exists in fact a proper inner analytic function $w_{p}(z)$ associated to

$$w_{p}^{-n\mbox{\Large$\cdot$}\!}(z) = w^{-n\mbox{\Large$\cdot$}\!}(z) - c_{0}^{(-n)},$$ (16)

namely its $n^{\rm th}$ angular derivative. As we have shown in [#!CAoRFI!#], from the $\rho\to 1_{(-)}$ limit to the unit circle of the real part $u^{-n\prime}(\rho,\theta)$ of the inner analytic function $w^{-n\mbox{\Large$\cdot$}\!}(z)$ we can recover the integrable real function $f^{-n\prime}(\theta)$, almost everywhere in its domain of definition. We thus have the correspondence for the integrable real function

$$f^{-n\prime}(\theta) \longleftrightarrow w^{-n\mbox{\Large$\cdot$}\!}(z).$$ (17)

Having done this, we now take the $n^{\rm th}$ angular derivative of the inner analytic function $w^{-n\mbox{\Large$\cdot$}\!}(z)$, thus obtaining a proper inner analytic function $w_{p}(z)=u_{p}(\rho,\theta)+\mbox{\boldmath$\imath$} v_{p}(\rho,\theta)$. Since $n$ angular differentiations of $w^{-n\mbox{\Large$\cdot$}\!}(z)$ correspond to $n$ differentiations with respect to $\theta$ of $f^{-n\prime}(\theta)$, and thus completely eliminates the piecewise polynomial real function $P_{(n-1)}(\theta)$ within the domain of definition of $f(\theta)$, it follows that the function $w_{p}(z)$ is a proper inner analytic function corresponding to the non-integrable real function $f(\theta)$. As discussed before, this correspondence is valid only up to a global real constant yet to be determined, so that we may now write

$$f(\theta) \longrightarrow \frac{\alpha_{0}}{2} + w_{p}(z),$$ (18)

where $\alpha_{0}$ can then be determined as discussed before, by the comparison between the real part $u_{p}(1,\theta)$ of $w_{p}(z)$ and $f(\theta)$ at some particular point on the unit circle where they do not have hard singularities. Having determined $\alpha_{0}$, and therefore $c_{0}=\alpha_{0}/2$, we may now define the inner analytic function that corresponds to $f(\theta)$,

$$w(z) = c_{0} + w_{p}(z),$$ (19)

so that we have the relation leading from $f(\theta)$ to $w(z)$

$$f(\theta) \longrightarrow w(z).$$ (20)

This completes the first part of the proof of Theorem 1.

Proof 1.2   :

We must now prove that we can recover $f(\theta)$ from the real part $u(\rho,\theta)$ of $w(z)$ in the $\rho\to 1_{(-)}$ limit. In order to do this, we start from the fact that from [#!CAoRFI!#] we know this to be true for the $n$-fold primitives

$$w^{-n\mbox{\Large$\cdot$}\!}(z) \longleftrightarrow f^{-n\prime}(\theta).$$ (21)

While the inner analytic function $w^{-n\mbox{\Large$\cdot$}\!}(z)$ is given by the power series

$$w^{-n\mbox{\Large$\cdot$}\!}(z) = \sum_{k=0}^{\infty} c_{k}^{(-n)}z^{k},$$ (22)

as we have shown in [#!CAoRFIII!#] the real function $f^{-n\prime}(\theta)$ can be expressed almost everywhere as an regulated Fourier series, even if the Fourier series itself diverges almost everywhere,

$$f^{-n\prime}(\theta) = \frac{\alpha_{0}^{(-n)}}{2} + \l... ...(-n)}\cos(k\theta) + \beta_{k}^{(-n)}\sin(k\theta) \right],$$ (23)

where we have that $c_{0}^{(-n)}=\alpha_{0}^{(-n)}/2$ and that $c_{k}^{(-n)}=\alpha_{k}^{(-n)}-\mbox{\boldmath$\imath$}\beta_{k}^{(-n)}$ for $k\in\{1,2,3,\ldots,\infty\}$. As we established in [#!CAoRFIII!#], since the sequences of Fourier coefficients $\alpha_{k}^{(-n)}$ and $\beta_{k}^{(-n)}$ are exponentially bounded, this series is absolutely and uniformly convergent for $0<\rho<1$. As we saw in [#!CAoRFI!#], the fact that we can recover $f^{-n\prime}(\theta)$ as the $\rho\to 1_{(-)}$ limit of the real part $u^{-n\prime}(\rho,\theta)$ of $w^{-n\mbox{\Large$\cdot$}\!}(z)$ is a consequence of the fact that the two real functions $u^{-n\prime}(1,\theta)$ and $f^{-n\prime}(\theta)$ have exactly the same set of Fourier coefficients. This, is turn, can be expressed as the relations between the Taylor coefficients $c_{k}^{(-n)}$ associated to $u(\rho,\theta)$ and the Fourier coefficients $\alpha_{k}^{(-n)}$ and $\beta_{k}^{(-n)}$ associated to $f(\theta)$, which have just been given above.

Let us now prove that the correspondence between $u^{-n\prime}(\rho,\theta)$ and $f^{-n\prime}(\theta)$ implies the same correspondence between $u^{(-n+1)\prime}(\rho,\theta)$ and $f^{(-n+1)\prime}(\theta)$, when we differentiate the two functions. We can do this by just showing that the relations between the Taylor coefficients and the Fourier coefficients are preserved by this process of differentiation. If we just differentiate the inner analytic function using angular differentiation we get

$$w^{(-n+1)\mbox{\Large$\cdot$}\!}(z) = \sum_{k=1}^{\infty} \mbox{\boldmath$\imath$}k\,c_{k}^{(-n)}z^{k}$$

$$= \sum_{k=1}^{\infty} c_{k}^{(-n+1)}z^{k},$$ (24)

from which we have for the Taylor coefficients $c_{k}^{(-n+1)}=\mbox{\boldmath$\imath$} k\,c_{k}^{(-n)}$, for $k\in\{1,2,3,\ldots,\infty\}$. Note that, since this is a convergent power series, we can always differentiate it term-by-term. If we now differentiate the real function using simple differentiation with respect to $\theta$ we get

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

$$= \lim_{\rho\to 1_{(-)}} \sum_{k=1}^{\infty} \rho^{k} \left[ \alpha_{k}^{(-n+1)}\cos(k\theta) + \beta_{k}^{(-n+1)}\sin(k\theta) \right],$$ (25)

from which we have for the corresponding Fourier coefficients that $\alpha_{k}^{(-n+1)}=k\,\beta_{k}^{(-n)}$ and also that $\beta_{k}^{(-n+1)}=-k\,\alpha_{k}^{(-n)}$, for $k\in\{1,2,3,\ldots,\infty\}$. Note that we have, in either case, that $c_{0}^{(-n+1)}=0$ and that $\alpha_{0}^{(-n+1)}=0$, so that the relation between $c_{0}^{(-n)}$ and $\alpha_{0}^{(-n)}$ is in fact preserved. Note also that, since this trigonometric series is uniformly convergent, and in fact is the real part of a convergent complex power series, we may differentiate it term-by-term so long as the series thus obtained is also convergent. Since the Fourier coefficients $\alpha_{0}^{(-n+1)}$ and $\beta_{0}^{(-n+1)}$, which increase at most as a power of $k$ when $k\to\infty$, are thus seen to be exponentially bounded, this implies that the series thus obtained is also absolutely and uniformly convergent, as we have shown in [#!CAoRFIII!#]. Therefore, we are justified in differentiating the original series term-by-term. We therefore have the relation between the Taylor coefficients $c_{k}^{(-n+1)}$ and the Fourier coefficients $\alpha_{k}^{(-n+1)}$ and $\beta_{k}^{(-n+1)}$,

$$c_{k}^{(-n+1)} = \mbox{\boldmath$\imath$}k\,c_{k}^{(-n)}$$

$$= \mbox{\boldmath$\imath$}k \left[ \alpha_{k}^{(-n)} - \mbox{\boldmath$\imath$} \beta_{k}^{(-n)} \right]$$

$$= \mbox{\boldmath$\imath$}k \left[ -\, \frac{1}{k}\, \beta_{k}^{(-n+1)} - \mbox{\boldmath$\imath$}\, \frac{1}{k}\, \alpha_{k}^{(-n+1)} \right]$$

$$= \alpha_{k}^{(-n+1)} - \mbox{\boldmath$\imath$} \beta_{k}^{(-n+1)}.$$ (26)

We see therefore that we indeed have that the relation between $c_{k}^{(-n)}$, $\alpha_{k}^{(-n)}$ and $\beta_{k}^{(-n)}$, for $k\in\{1,2,3,\ldots,\infty\}$, is also preserved,

$$c_{k}^{(-n+1)} = \alpha_{k}^{(-n+1)} - \mbox{\boldmath$\imath$} \beta_{k}^{(-n+1)},$$ (27)

which thus establishes the correspondence for the first derivatives. We may now extend this argument to subsequent derivatives, from $w^{(-n+1)\mbox{\Large$\cdot$}\!}(z)$ and $f^{(-n+1)\prime}(\theta)$ all the way to $w(z)$ and $f(\theta)$, by finite induction. Therefore, let us assume the result for the case $(-n+i)$ and show that this implies that it is also valid for the case $(-n+i+1)$. We assume therefore that we have

$$c_{k}^{(-n+i)} = \alpha_{k}^{(-n+i)} - \mbox{\boldmath$\imath$} \beta_{k}^{(-n+i)},$$ (28)

for some positive value of $i$, where the functions are expressed as the corresponding series

$$w^{(-n+i)\mbox{\Large$\cdot$}\!}(z) = \sum_{k=1}^{\infty} c_{k}^{(-n+i)}z^{k},$$

$$f^{(-n+i)\prime}(\theta) = \lim_{\rho\to 1_{(-)}} \sum_{k=1}^{\infty} \rho^{k} \left[ \alpha_{k}^{(-n+i)}\cos(k\theta) + \beta_{k}^{(-n+i)}\sin(k\theta) \right].$$ (29)

We may now differentiate either series term-by-term, which we may do for the same reasons as before, thus obtaining

$$w^{(-n+i+1)\mbox{\Large$\cdot$}\!}(z) = \sum_{k=1}^{\infty} \mbox{\boldmath$\imath$}k\,c_{k}^{(-n+i)}z^{k},$$

$$f^{(-n+i+1)\prime}(\theta) = \lim_{\rho\to 1_{(-)}} \sum_{k=1}^{\infty} \rho^{k} \left[ k\,\beta_{k}^{(-n+i)}\cos(k\theta) - k\,\alpha_{k}^{(-n+i)}\sin(k\theta) \right],$$ (30)

so that we have for the coefficients for the case $(-n+i+1)$

$$c_{k}^{(-n+i+1)} = \mbox{\boldmath$\imath$}k\,c_{k}^{(-n+i)},$$

$$\alpha_{k}^{(-n+i+1)} = k\,\beta_{k}^{(-n+i)},$$

$$\beta_{k}^{(-n+i+1)} = - k\,\alpha_{k}^{(-n+i)},$$ (31)

for $k\in\{1,2,3,\ldots,\infty\}$. Using now the relations between the coefficients for the case $(-n+i)$ we have

$$c_{k}^{(-n+i+1)} = \mbox{\boldmath$\imath$}k\,c_{k}^{(-n+i)}$$

$$= \mbox{\boldmath$\imath$}k \left[ \alpha_{k}^{(-n+i)} - \mbox{\boldmath$\imath$} \beta_{k}^{(-n+i)} \right]$$

$$= \mbox{\boldmath$\imath$}k \left[ -\, \frac{1}{k}\, \beta_{k}^{(-n+i+1)} - \mbox{\boldmath$\imath$}\, \frac{1}{k}\, \alpha_{k}^{(-n+i+1)} \right]$$

$$= \alpha_{k}^{(-n+i+1)} - \mbox{\boldmath$\imath$} \beta_{k}^{(-n+i+1)},$$ (32)

for $k\in\{1,2,3,\ldots,\infty\}$, thus showing that the relation between the coefficients is in fact preserved, where we recall that the $k=0$ coefficients are always zero during this process. This is therefore true for all possible multiple derivatives, all the way to infinity, and in particular it is true for $i=n$, that is, for the coefficients of $w(z)$ and $f(\theta)$. In order to complete the proof in this case all we have to do is to consider the real function

$$g(\theta) = u(1,\theta)-f(\theta),$$ (33)

where

$$u(1,\theta) = \lim_{\rho\to 1_{(-)}} u(\rho,\theta).$$ (34)

Since the expression of the Fourier coefficients is linear on the functions, and since $u(1,\theta)$ and $f(\theta)$ have exactly the same set of Fourier coefficients, it is clear that all the Fourier coefficients of $g(\theta)$ are zero. Therefore, for the real function $g(\theta)$ we have that $c_{k}=0$ for all $k$, and thus the inner analytic function that corresponds to $g(\theta)$ is the identically null complex function $w_{\gamma}(z)\equiv 0$. This is an inner analytic function which is, in fact, analytic over the whole complex plane, and which, in particular, is zero over the unit circle, so that we have $g(\theta)\equiv 0$, since the identically zero real function is the only integrable real function without removable singularities that corresponds to the identically zero inner analytic function, due to the completeness of the Fourier basis, as was shown in [#!CAoRFIII!#]. Note, in particular, that the $\rho\to 1_{(-)}$ limits exist at all points of the unit circle in the case of the inner analytic function associated to $g(\theta)$. Since both $u(1,\theta)$ and $f(\theta)$ have non-integrable hard singularities at isolated points on the unit circle, we can conclude only that

$$f(\theta) = \lim_{\rho\to 1_{(-)}} u(\rho,\theta)$$ (35)

almost everywhere on the unit circle, or everywhere in the domain of definition of $f(\theta)$, which therefore excludes all the points where the function has hard singularities. Note that the domain of definition of $u(1,\theta)$ is the same as that of $f(\theta)$, because any hard singularities that may have been softened in the process of iterative integration, during the construction of $w(z)$, will have been hardened again in the corresponding process of iterative differentiation. We have therefore the complete correspondence

$$f(\theta) \longleftrightarrow w(z).$$ (36)

The inner analytic function $w(z)$ represents the non-integrable real function $f(\theta)$ exactly in the same way as that which was established for integrable real functions in [#!CAoRFI!#]. Note that this proof establishes that the correspondence between the inner analytic functions and the real functions is also valid for all the intermediate cases, from $w^{-n\mbox{\Large$\cdot$}\!}(z)$ and $f^{-n\prime}(\theta)$ to $w(z)$ and $f(\theta)$. This completes the second part of the proof of Theorem 1.

Let us once more draw attention to the fact that the inner analytic function $w(z)$ produced in the way described above, while it does correspond to the non-integrable real function $f(\theta)$, is not unique. One can add to it any linear combination of inner analytic functions that correspond to singular distributions with their singularities at the singular points on the unit circle corresponding to the angles $\{\theta_{1},\ldots,\theta_{N}\}$, without changing the fact that the non-integrable real function $f(\theta)$ is still recovered as the $\rho\to 1_{(-)}$ limit of the real part of the new inner analytic function obtained in this way, at all points where it is well defined.

Given the inner analytic function $w(z)$ that was obtained from $f(\theta)$ by the process described above, one can then consider defining from it a reduced inner analytic function $w_{r}(z)$ that represents the non-integrable real function in a unique way, without the superposition of any singular distributions. If the $N$ singular points of $w(z)$ are examined in order to determine the existence there of singular distributions, and since each one of these singular distributions is represented by a known unique inner analytic function of its own, as was shown in [#!CAoRFII!#], one can simply subtract from $w(z)$ the appropriate linear combination of inner analytic functions of the singular distributions present, in order to obtain an inner analytic function that represents the non-integrable real function $f(\theta)$ alone, without any singular distributions superposed to it.

The simplest way to do this is to examine the result of every angular differentiation during the process leading from $w^{-n\mbox{\Large$\cdot$}\!}(z)$ to $w(z)$. At each step one can verify whether or not the angular differentiation has generated one or more Dirac delta ``functions'' at some of the singular points. This is a simple thing to verify, because the occurrence of a delta ``function'' at a certain point is always preceded by the occurrence of a finite discontinuity of the real function at that point. This can also be done by the determination of the type and orientation of the singularities at these points, since the Dirac delta ``functions'' are associated to inner analytic functions with simple poles that have a specific orientation with respect to the unit circle. One can then subtract from the proper inner analytic function obtained at that point in the iterative differentiation process the inner analytic functions corresponding to the delta ``functions'' at the appropriate points. By doing this one guarantees that no derivatives of delta ``functions'' will ever arise during the process of multiple differentiation. After one determines $\alpha_{0}$ at the last step of the process, this will lead to a reduced inner analytic function $w_{r}(z)$ which includes no singular distributions at all, and that hence corresponds to $f(\theta)$ in a unique and simple way,

$$f(\theta) \longleftrightarrow w_{r}(z),$$ (37)

that does not include the superposition of any singular distributions with support on the singular points between successive sections.