If we have a holomorphic function f defined on the open unit disk \mathbb{D}=\{z:|z|, it is reasonable to ask under what conditions we can extend this function to the boundary of the unit disk. To do this, we can look at what the function looks like on each circle inside the disk centered at 0, each with some radius r. This defines a new function: :\begin{cases} f_r:S^1 \to \Complex \\ f_{r}(e^{i\theta})=f(re^{i\theta}) \end{cases} where :S^1:=\{e^{i\theta}:\theta\in[0,2\pi]\}=\{z\in \Complex:|z|=1\}, is the
unit circle. Then it would be expected that the values of the extension of f onto the circle should be the limit of these functions, and so the question reduces to determining when f_r converges, and in what sense, as r\to 1, and how well defined is this limit. In particular, if the
L^p norms of these f_r are well behaved, we have an answer: :
Theorem. Let f:\mathbb{D}\to\Complex be a holomorphic function such that :: \sup_{0 :where f_r are defined as above. Then f_r converges to some function f_1\in L^p(S^{1})
pointwise almost everywhere and in L^p norm. That is, ::\begin{align} \left |f_r(e^{i\theta})-f_{1}(e^{i\theta}) \right | &\to 0 && \text{for almost every } \theta\in [0,2\pi] \\ \|f_r-f_1\|_{L^p(S^1)} &\to 0 \end{align} Now, notice that this pointwise limit is a radial limit. That is, the limit being taken is along a straight line from the center of the disk to the boundary of the circle, and the statement above hence says that : f(re^{i\theta})\to f_1(e^{i\theta}) \qquad \text{for almost every } \theta. The natural question is, with this boundary function defined, will we converge pointwise to this function by taking a limit in any other way? That is, suppose instead of following a straight line to the boundary, we follow an arbitrary curve \gamma:[0,1)\to \mathbb{D} converging to some point e^{i\theta} on the boundary. Will f converge to f_{1}(e^{i\theta})? (Note that the above theorem is just the special case of \gamma(t)=te^{i\theta}). It turns out that the curve \gamma needs to be
non-tangential, meaning that the curve does not approach its target on the boundary in a way that makes it tangent to the boundary of the circle. In other words, the range of \gamma must be contained in a wedge emanating from the
limit point. We summarize as follows:
Definition. Let \gamma:[0,1)\to \mathbb{D} be a continuous path such that \lim\nolimits_{t\to 1}\gamma(t)=e^{i\theta}\in S^{1}. Define : \begin{align} \Gamma_\alpha &=\{z:\arg z\in [\pi-\alpha,\pi+\alpha]\} \\ \Gamma_\alpha(\theta) &=\mathbb{D}\cap e^{i\theta}(\Gamma_\alpha+1) \end{align} That is, \Gamma_\alpha(\theta) is the wedge inside the disk with angle 2\alpha whose axis passes between e^{i\theta} and zero. We say that \gamma converges
non-tangentially to e^{i\theta}, or that it is a
non-tangential limit, if there exists 0 such that \gamma is contained in \Gamma_\alpha(\theta) and \lim\nolimits_{t\to 1}\gamma(t)=e^{i\theta}. :'''Fatou's Theorem.''' Let f\in H^p(\mathbb{D}). Then for almost all \theta\in[0,2\pi], ::\lim_{t\to 1}f(\gamma(t))=f_1(e^{i\theta}) :for every non-tangential limit \gamma converging to e^{i\theta}, where f_1 is defined as above. Here, H^p(\mathbb{D}) is the
Hardy space. ==Discussion==