Conclusions

Two very weak conditions were established leading to the existence of corresponding inner analytic functions, and thus to the representability of real functions or other objects defined on the unit circle by their sequence of Fourier coefficients. One of them is a condition on the sequence of coefficients, the other is an analytic condition on the real functions. The first one, stated in the most general way possible, reads:





% latex2html id marker 2141
$\textstyle \parbox{\textwidth}{\bf\boldmath If a s...
... from
them converges to an inner analytic function inside the open unit disk.}$





This is a very weak condition on the sequence of coefficients, leading to the representability of the object related to it by an inner analytic function. The object at issue may be an integrable real function, or it may be a singular object that has an integrability concept associated to it. The typical example of such singular objects is Dirac's delta ``function'' and the Schwartz distribution associated to it. This is therefore a very general condition, which in fact extrapolates the strict realm of real functions. The second one reads:





$\textstyle \parbox{\textwidth}{\bf\boldmath Any Lebesgue-measurable integrable ...
...val, regardless of whether or not the corresponding Fourier
series converges.}$





This is a very weak analytical condition on the real functions, leading to the representability of each real function by a corresponding inner analytic function. Observe that, so long as the real functions at issue are Lebesgue-measurable, this is the statement that every integrable function is representable by its sequence of Fourier coefficients. All that is really needed is that these Fourier coefficients, and hence the integrals that give them, exist.

If one accepts the premise that the condition of integrability is necessary for the very existence of the sequence of Fourier coefficients, and since it is seen here to be also sufficient for the representability of the function by those coefficients, we may say that what we have here is in essence a necessary and sufficient condition for the representability of real functions by their Fourier coefficients, as defined by the usual integrals. This is similar in nature to the as yet open problem of establishing a necessary and sufficient condition for the convergence of the Fourier series of a real function. If we reinterpret this last one as a condition for the representability of the function by its series, we see that such a necessary and sufficient condition can be achieved if one exchanges the condition of representability by the series for a more general one, involving the representability by the sequence of coefficients, and the recovery of the real function from them via the construction of an analytic function within the open unit disk.

However, it is important to note that, although this condition is always sufficient, it is only necessary if one assumes that the Fourier coefficients must be given by the usual integrals over the periodic interval. As we will see elsewhere, there is in fact a large class of non-integrable real functions for which a set of Fourier coefficients can still be consistently defined, and which can still be represented by this set of Fourier coefficients, almost everywhere over the unit circle.

Although the process of recovery of the real function as a limit of the corresponding inner analytic function is, by itself, not algorithmic in nature, it can lead to algorithmic solutions in at least some cases. The results obtained here simply guarantee the existence of the inner analytic function, and at least in principle the possibility of the recovery of the real function through the limit of that inner analytic function to the unit circle. In sufficiently simple cases, namely when the inner analytic function has only a finite number of sufficiently soft known singularities on the unit circle, it is possible to devise expressions involving modified trigonometric series that converge to the function, as was shown in [2]. In this case, one acquires an algorithmic method for the calculation of the real function to any desired level of precision, and thus for the practical recovery of the real function from the Fourier coefficients, which can be made good enough for any practical purpose.