Review of Real Functions

When we discuss real functions in this paper, some properties will be globally assumed for these functions, just as was done in [#!CAoRFI!#,#!CAoRFII!#,#!CAoRFIII!#]. These are rather weak conditions to be imposed on these functions, that will be in force throughout this paper. It is to be understood, without any need for further comment, that these conditions are valid whenever real functions appear in the arguments. These weak conditions certainly hold for any integrable real functions that are obtained as restrictions of corresponding inner analytic functions to the unit circle.

The most basic condition is that the real functions must be measurable in the sense of Lebesgue, with the usual Lebesgue measure [#!RARudin!#,#!RARoyden!#]. The second global condition we will impose is that the functions have no removable singularities. The third and last global condition is that the number of hard singularities on the unit circle be finite, and hence that they be all isolated from one another. There will be no limitation on the number of soft singularities.

In addition to this we will assume, for the purposes of this particular paper, that all real functions are integrable on the unit circle and, just for the sake of clarity and simplicity, unless explicitly stated otherwise we will also assume that all real functions are zero-average real functions, meaning that their integrals over the unit circle are zero. Since this simply implies that the Fourier coefficients $\alpha_{0}$ of the real functions are zero, without affecting any of the other coefficients in any way, this clearly has no impact on any arguments about the convergence of the Fourier series.

For the purposes of this paper it is important to review here, in some detail, the construction that results in the correspondence between integrable real functions on the unit circle and inner analytic functions on the open unit disk. In [#!CAoRFI!#] we showed that, given any integrable real function $f(\theta)$, one can construct a corresponding inner analytic function $w(z)$, 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!#] $\alpha_{k}$ and $\beta_{k}$ 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 $c_{k}$ given by

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

$$c_{k} = \alpha_{k} - \mbox{\boldmath$\imath$}\beta_{k},$$ (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 complex power series

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

which we showed in [#!CAoRFI!#] to be convergent to an inner analytic function $w(z)$ within the open unit disk. That inner analytic function may be written as

$$w(z) = u(\rho,\theta) + \mbox{\boldmath$\imath$} v(\rho,\theta).$$ (7)

The complex power series in Equation (6) is therefore the Taylor series of $w(z)$. We also proved in [#!CAoRFI!#] that one recovers the real function $f(\theta)$ almost everywhere on the unit circle from the real part $u(\rho,\theta)$ of $w(z)$, by means of the $\rho\to 1_{(-)}$ limit. The $\rho\to 1_{(-)}$ limit of the imaginary part $v(\rho,\theta)$ also exists almost everywhere and gives rise to a real function $g(\theta)$ which corresponds to $f(\theta)$. The pair of real functions obtained from the real and imaginary parts of one and the same inner analytic function are said to be mutually Fourier-conjugate real functions.

In a subsequent paper [#!CAoRFIII!#] we showed that all the elements of the Fourier theory [#!FSchurchill!#] of integrable real functions are contained within the complex-analytic structure, including the Fourier basis of functions, the Fourier series, the scalar product for integrable real functions, the relations of orthogonality and norm of the basis elements, and the completeness of the Fourier basis, including its so-called completeness relation. As was also shown in [#!CAoRFIII!#] the real function $g(\theta)=v(1,\theta)$ which is the Fourier-conjugate function to $f(\theta)=u(1,\theta)$ has the same Fourier coefficients, but with the meanings of $\alpha_{k}$ and $\beta_{k}$ interchanged in such a way that we have

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

$$\beta_{k} = -\, \frac{1}{\pi} \int_{-\pi}^{\pi}d\theta\, \cos(k\theta) g(\theta),$$ (8)

for $k\in\{1,2,3,\ldots,\infty\}$. Note that there is no constant term in the Fourier series of $g(\theta)$, which means that we have

$$\int_{-\pi}^{\pi}d\theta\, g(\theta) = 0.$$ (9)

In other words, the Fourier-conjugate function $g(\theta)$ is always a zero-average real function. Note also that this fact, as well as the relations in Equation (8) imply, in particular, that $g(\theta)$ is also an integrable real function. We may therefore conclude that, if $f(\theta)$ is an integrable real function, then so is its Fourier-conjugate function $g(\theta)$.