Let be a holomorphic function. What conditions do we have to impose on on the boundary to guarantee convergence (in some sense) to boundary values? We begin by formulating a suitable form of convergence to answer this question.
We say that a function has a radial limit at the point on the circle if the limit
Before we give sufficient conditions, we need to review some basic results about Fourier series. For and , we have by the geometric series formula the following identity:
We call the Poisson kernel on the unit disk .
Proof. Noting that, for , , we obtain from the geometric series formula that
We use the Poisson kernel to prove the next theorem on the convergence of Fourier series. We will use (without proof of) the result that is an approximate identity.
Theorem 1. Suppose . Denote the Fourier coefficients of by , .
- If for all , then a.e.
- for a.e. , as .
Proof. (1) is a consequence of (2), so we only show the latter. Changing to if necessary, we may assume without loss of generality that , so that has a clear -periodic extension to all of . By the dominated convergence theorem and the above identity,
where the penultimate equality follows from translation invariance and periodicity. Since the Poisson kernel is an approximate identity,
at every Lebesgue point of , hence almost everywhere.
As above, let denote the Fourier coefficient of a function . We now prove some fundamental results for Fourier series of functions in .
Theorem 2. Suppose .
- (Parseval’s Identity)
- The mapping defines a unitary transformation from to .
- , where .
Proof. Assertions (1), (2), and (3) follow from more general Hilbert space results for complete orthonormal bases. We know that is an orthonormal system, so it remains for us to completeness. Recall that a system is complete if and only if the only element orthogonal to all the elements is . By assertion (1) of Theorem 1,
implies that a.e.
Theorem 3. (Fatou) A bounded holomorphic function has radial limits at almost every .
Proof. Since is holomorphic on , we can write , for , where convergence holds absolutely uniformly for with . For , define a curve by . By the uniqueness of Laurent series and the formula for Laurent coefficients,
Note that the last integral vanishes for , since is holomorphic on . Hence, is the Fourier series of the function , for fixed.
Let be such that for all . By Parseval’s identity,
Since is dominated by , hence , it follows from the dominated convergence theorem that . Let denote the function whose Fourier coefficients , . By assertion (2) of Theorem 1 above,
for a.e. , which completes the proof of the theorem.
We define the Hardy space to be the set of all holomorphic functions such that
It follows from Minkowski’s inequality and basic limit properties that
defines a norm on . We will later see that we can extend the definition of Hardy spaces on the unit disk to arbitrary .
Stay tuned for the second installment on Fatou’s theorem!