Representation in the Extended Fourier Theory

Assuming that, given a non-integrable real function $f(\theta)$, the corresponding reduced inner analytic function $w_{r}(z)=u_{r}(\rho,\theta)+\mbox{\boldmath$\imath$}v_{r}(\rho,\theta)$ has been determined, we may now consider determining a unique set of Fourier coefficients to be associated to the non-integrable real function $f(\theta)$. Of these, $\alpha_{0}$ has already been determined, through the comparison of $u_{r}(1,\theta)$ and $f(\theta)$ at some point of the unit circle where they do not have hard singularities. It follows that $c_{0}=\alpha_{0}/2$ has also been determined. From the Taylor series of $w_{r}(z)$,

$$w_{r}(z) = \sum_{k=0}^{\infty} c_{k}z^{k},$$ (38)

we have the values of all the other Taylor coefficients $c_{k}$, for $k\in\{1,2,3,\ldots,\infty\}$. Given that in this case we have that $c_{k}=\alpha_{k}-\mbox{\boldmath$\imath$}\beta_{k}$, we immediately get the values of all the other Fourier coefficients $\alpha_{k}$ and $\beta_{k}$. Note that this construction has the effect of associating a complete set of Fourier coefficients $\alpha_{0}$, $\alpha_{k}$ and $\beta_{k}$, for $k\in\{1,2,3,\ldots,\infty\}$, to the non-integrable real function $f(\theta)$. In particular, the association of $\alpha_{0}$ has the effect of attributing an average value to the non-integrable real function $f(\theta)$, which has been defined via an analytic process.

Although these coefficients obviously cannot be written in terms of integrals involving $f(\theta)$ in the usual way, all of them except for $\alpha_{0}$ can, in fact, be written as a certain set of integrals. In order to do this, we start by considering the $n^{\rm th}$ angular primitive $w_{r}^{-n\mbox{\Large$\cdot$}\!}(z)$ of the reduced inner analytic function $w_{r}(z)$, which is, therefore, a proper inner analytic function, and the known associated Fourier and Taylor coefficients, which we will name $\alpha_{0}^{(-n)}$, $\alpha_{k}^{(-n)}$ and $\beta_{k}^{(-n)}$, for $k\in\{1,2,3,\ldots,\infty\}$, and $c_{k}^{(-n)}$, for $k\in\{0,1,2,3,\ldots,\infty\}$. Since it has at most borderline hard singularities on the unit circle, this inner analytic function corresponds to an integrable real function $f_{r}^{-n\prime}(\theta)$ on that circle,

$$f_{r}^{-n\prime}(\theta) = \lim_{\rho\to 1_{(-)}} u_{r}^{-n\mbox{\Large$\cdot$}\!}(\rho,\theta),$$ (39)

so that the Fourier coefficients of $f_{r}^{-n\prime}(\theta)$ can be written as integrals involving this function,

$$\alpha_{k}^{(-n)} = \frac{1}{\pi} \int_{-\pi}^{\pi}d\theta\, \cos(k\theta)f_{r}^{-n\prime}(\theta),$$

$$\beta_{k}^{(-n)} = \frac{1}{\pi} \int_{-\pi}^{\pi}d\theta\, \sin(k\theta)f_{r}^{-n\prime}(\theta),$$ (40)

for $k\in\{1,2,3,\ldots,\infty\}$. We now recall that we have for the Taylor series of the $n^{\rm th}$ angular primitive of $w_{r}(z)$,

$$w_{r}^{-n\mbox{\Large$\cdot$}\!}(z) = \sum_{k=1}^{\infty} c_{k}^{(-n)}z^{k},$$ (41)

and therefore the $n^{\rm th}$ angular derivative of this equation is given by

$$w_{r}^{0\mbox{\Large$\cdot$}\!}(z) = \sum_{k=1}^{\infty} ... ... \mbox{\boldmath$\imath$}^{n}k^{n}c_{k}^{(-n)} \right]z^{k},$$ (42)

where $w_{r}^{0\mbox{\Large$\cdot$}\!}(z)$ is the proper inner analytic function associated to $w_{r}(z)$. It thus follows that we have for the Taylor coefficients $c_{k}$ of $w_{r}^{0\mbox{\Large$\cdot$}\!}(z)$, which are also the Taylor coefficients of $w_{r}(z)$, for $k\in\{1,2,3,\ldots,\infty\}$,

$$c_{k} = \mbox{\boldmath$\imath$}^{n}k^{n}c_{k}^{(-n)}.$$ (43)

Since we know that the coefficients $c_{k}^{(-n)}$, being the Taylor coefficients associated to an integrable real function, are limited for all $k$, we may now conclude that the coefficients $c_{k}$ of the non-integrable real function may diverge with $k$, but not faster than the power $k^{n}$. We may write this relation in terms of the Fourier coefficients $\alpha_{k}$ and $\beta_{k}$ associated to the non-integrable real function $f(\theta)$, for $k\in\{1,2,3,\ldots,\infty\}$, if we recall from [#!CAoRFI!#] that $c_{k}=\alpha_{k}-\mbox{\boldmath$\imath$}\beta_{k}$, and also that $c_{k}^{(-n)}=\alpha_{k}^{(-n)}-\mbox{\boldmath$\imath$}\beta_{k}^{(-n)}$,

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

$$= \mbox{\boldmath$\imath$}^{n}k^{n} \alpha_{k}^{(-n)} - \mbox{\boldmath$\imath$}^{n+1}k^{n} \beta_{k}^{(-n)}.$$ (44)

We now see that the relations between the Fourier coefficients $(\alpha_{k},\beta_{k})$ and the Fourier coefficients $\left[\alpha_{k}^{(-n)},\beta_{k}^{(-n)}\right]$ depend on the parity of $n$. For even $n=2j$ we have

$$\alpha_{k} = (-1)^{j}k^{n} \alpha_{k}^{(-n)},$$

$$\beta_{k} = (-1)^{j}k^{n} \beta_{k}^{(-n)},$$ (45)

while for odd $n=2j+1$ we have

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

$$\beta_{k} = (-1)^{j+1}k^{n} \alpha_{k}^{(-n)}.$$ (46)

Since we have the coefficients $\alpha_{k}^{(-n)}$ and $\beta_{k}^{(-n)}$ written as integrals, we may now write $\alpha_{k}$ and $\beta_{k}$ as integrals, first for the case of even $n=2j$,

$$\alpha_{k} = \frac{(-1)^{j}k^{n}}{\pi} \int_{-\pi}^{\pi}d\theta\, \cos(k\theta)f_{r}^{-n\prime}(\theta),$$

$$\beta_{k} = \frac{(-1)^{j}k^{n}}{\pi} \int_{-\pi}^{\pi}d\theta\, \sin(k\theta)f_{r}^{-n\prime}(\theta),$$ (47)

and then for the case of odd $n=2j+1$,

$$\alpha_{k} = \frac{(-1)^{j}k^{n}}{\pi} \int_{-\pi}^{\pi}d\theta\, \sin(k\theta)f_{r}^{-n\prime}(\theta),$$

$$\beta_{k} = \frac{(-1)^{j+1}k^{n}}{\pi} \int_{-\pi}^{\pi}d\theta\, \cos(k\theta)f_{r}^{-n\prime}(\theta).$$ (48)

Since the integrals shown are all limited as functions of $k$, given that $f_{r}^{-n\prime}(\theta)$ is an integrable real function, once again it is apparent that the coefficients $\alpha_{k}$ and $\beta_{k}$ which are associated to $f_{r}(\theta)$ typically diverge as a positive power of $k$ when $k\to\infty$. Note that, when the real function $f(\theta)$ is integrable on the whole unit circle, we are reduced to the case $n=0$, so that the expressions for the even case reduce to the usual ones for the Fourier coefficients.

In a previous paper [#!CAoRFIII!#] we showed that the whole Fourier theory of integrable real functions is contained in the complex-analytic structure introduced in [#!CAoRFI!#]. We also extended that Fourier theory to include, not only the singular distributions discussed in [#!CAoRFII!#], but essentially the whole space of inner analytic functions. We now observe that the sequences of complex Taylor coefficients $c_{k}$ in Equation (43), which go to infinity with $k$ not faster than a power, are exponentially bounded, according to the definition of that term given in [#!CAoRFIII!#], and repeated in Equation (7). This in itself suffices to show that the corresponding power series converges to an inner analytic function, as was shown in that paper. It also implies, as was also shown there, that the two sequences of real coefficients $\alpha_{k}$ and $\beta_{k}$ associated to $f(\theta)$ are both also exponentially bounded. As a consequence of all this, the non-integrable real functions we are discussing in this paper can be expressed as regulated Fourier series, as given in Equation (8), thus using the summation rule that was presented in [#!CAoRFIII!#],

$$f(\theta) = \frac{\alpha_{0}}{2} + \lim_{\rho\to 1_{(-)... ... \alpha_{k}\cos(k\theta) + \beta_{k}\sin(k\theta) \right],$$ (49)

which is equivalent to the fact that the non-integrable real functions which we discussed here can be obtained as the $\rho\to 1_{(-)}$ limits of the real parts of inner analytic functions. We see therefore that the class of non-integrable real functions which we are examining here is also contained in the extended Fourier theory presented in [#!CAoRFIII!#].