Fatou's theorem
Encyclopedia
In complex analysis
, Fatou's theorem, named after Pierre Fatou
, is a statement concerning holomorphic functions on the unit disk and their pointwise extension to the boundary of the disk.
defined on the open unit disk 
, 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 
. This defines a new function on the circle 
, defined by 
, where 
. Then it would be expected that the values of the extension of 
onto the circle should be the limit of these functions, and so the question reduces to determining when 
converges, and in what sense, as 
, and how well defined is this limit. In particular, if the L-p norms of these 
are well behaved, we have an answer:
Then
converges to some function 
pointwise
almost everywhere
and in
. That is,
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
for almost every
. The natural question is, now 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 
converging to some point 
on the boundary. Will 
converge to 
? (Note that the above theorem is just the special case of 
).
It turns out that the curve
needs to be nontangential, 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 
must be contained in a wedge emanating from the limit point. We summarize as follows:
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
, Fatou's theorem, named after Pierre Fatou
Pierre Fatou
Pierre Joseph Louis Fatou was a French mathematician working in the field of complex analytic dynamics. He entered the École Normale Supérieure in Paris in 1898 to study mathematics and graduated in 1901 when he was appointed an astronomy post in the Paris Observatory...
, is a statement concerning holomorphic functions on the unit disk and their pointwise extension to the boundary of the disk.
Motivation and statement of theorem
If we have a holomorphic function defined on the open unit disk , 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 . This defines a new function on the circle , defined by , where . Then it would be expected that the values of the extension of onto the circle should be the limit of these functions, and so the question reduces to determining when converges, and in what sense, as , and how well defined is this limit. In particular, if the L-p norms of these are well behaved, we have an answer:- Theorem: Let
be a holomorphic function such that
Then
converges to some function pointwisePointwise
In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f of some function f. An important class of pointwise concepts are the pointwise operations — operations defined on functions by applying the operations to function values...
almost everywhere
Almost everywhere
In measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...
and in
. That is,
-
-
- for almost every
.
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
for almost every
. The natural question is, now 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 converging to some point on the boundary. Will converge to ? (Note that the above theorem is just the special case of ).It turns out that the curve
needs to be nontangential, 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 must be contained in a wedge emanating from the limit point. We summarize as follows:- Definition: Let
be a continuous path such that 
. Define
- That is,
is the wedge inside the disk with angle 
: whose axis passes between 
and zero. We say that 
- converges nontangentially to
, or that it is a nontangential limit, : if there exists 
such that 
is contained in 
and 
.
- Fatou's theorem: Let
. Then for almost all 
, 
- for every nontangential limit
converging to 
, where 
is defined as above.
Discussion
- The proof utilizes the symmetry of the Poisson kernelPoisson kernelIn potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disc. The kernel can be understood as the derivative of the Green's function for the Laplace equation...
using the Hardy–Littlewood maximal function for the circle. - The analogous theorem is frequently defined for the Hardy space over the upper-half plane and is proved in much the same way.