Extended Fourier Theory

In [#!CAoRFIII!#] we showed that the whole Fourier theory of integrable real functions is contained within the complex-analytic structure presented in [#!CAoRFI!#]. There we also established an extension of the Fourier theory to essentially all inner analytic functions. In particular, we introduced the concept of an exponentially bounded sequence of either Taylor or Fourier coefficients. Here is that definition: 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

$$\lim_{k\to\infty} \vert a_{k}\vert\,{\rm e}^{-Ck} = 0,$$ (7)

for all real $C>0$, then we say that the sequence of coefficients $a_{k}$ is exponentially bounded. This definition applies equally well to either complex Taylor coefficients or to real Fourier coefficients. In the construction of the inner analytic functions presented in [#!CAoRFI!#], if $\alpha_{0}$, $\alpha_{k}$ and $\beta_{k}$, for $k\in\{1,2,3,\ldots,\infty\}$, are the Fourier coefficients of an integrable real function $f(\theta)$, then we construct from them the set of complex Taylor coefficients $c_{0}=\alpha_{0}/2$ and $c_{k}=\alpha_{k}-\mbox{\boldmath$\imath$}\beta_{k}$, for $k\in\{1,2,3,\ldots,\infty\}$. We also showed in [#!CAoRFIII!#] that the condition that the sequence of Taylor coefficients $c_{k}$ be exponentially bounded is equivalent to the condition that the corresponding sequences of Fourier coefficients $\alpha_{k}$ and $\beta_{k}$ be both exponentially bounded. Another result that was obtained in [#!CAoRFIII!#] is that the condition that the sequence of Taylor coefficients $c_{k}$ be exponentially bounded is sufficient to ensure that the corresponding power series is convergent within the open unit disk, and therefore converges to an inner analytic function. Finally, we showed in that paper that the condition that the Fourier coefficients be exponentially bounded suffices to guarantee that the regulated Fourier series given by

$$u(\rho,\theta) = \frac{\alpha_{0}}{2} + \sum_{k=1}^{\in... ...[ \alpha_{k}\cos(k\theta) + \beta_{k}\sin(k\theta) \right]$$ (8)

converges absolutely and uniformly for $0<\rho<1$, and that $f(\theta)$ can be obtained as the $\rho\to 1_{(-)}$ limit of this regulated Fourier series almost everywhere on the unit circle,

$$f(\theta) = \frac{\alpha_{0}}{2} + \lim_{\rho\to 1_{(-)... ... \alpha_{k}\cos(k\theta) + \beta_{k}\sin(k\theta) \right].$$ (9)

This has the effect of extending the applicability of the Fourier theory far beyond the realm of integrable real functions, including, for example, the singular Schwartz distributions.