6.3 Analytic Capacity
The theory of rational approximation is essentially built upon the concept of analytic capacity, which was introduced in 1940 by Finnish mathematician Lars Ahlfors. Our purpose here is to give a brief and elementary introduction. Despite its importance, still many trivially simple results remain conjecture.
The uses of analytic capacity are present in many other topics of complex analysis. Analytic capacity serves as a natural framework for general rational approximation theory. Our purpose here is to hint at how analytic capacity theory relates to the proof of Theorem 6.2.2 and pertinent problems in general.
Definition 6.3.1 (Analytic Capacity).
Let be compact. The analytic capacity of is defined as
where is defined as in (6.3.1). For an arbitrary set , we define
Intuitively, measures the extent to which bounded analytic functions outside can deviate from constancy. Generally, the “larger” is, the greater the capacity is.
Proposition 6.3.1.
If is a compact set of discrete points, then .
Proof.
For any holomorphic with , since is bounded, the Riemann’s Theorem for removable singularities (Theorem 3.2.6) allows for an analytic continuation onto all of . Then Liouville’s Theorem (Theorem 3.2.3) implies that is constant and . Hence .
Theorem 6.3.1.
For both compact in , .
Proof.
This follows directly from the definition and the fact that any function holomorphic on is also holomorphic on .
The preceding results above hint at the monotonous behavior of capacity. However, currently it is not known whether a general subadditivity property holds for analytic capacity, or that
Recent results hint the affirmative, as many special cases of the relation have been proved; the question of subadditivity has been proved in the affirmative for disjoint compact continua, and recent findings by Xavier Tolsa show that capacity is (countably) semi-(sub)additive (the existence of an absolute constant such that ).
We now give some quantifying examples of how analytic capacity measures a type of “largeness” of compact sets, (rather much like area, which satisfies the subadditivity relation). First we define a specific classification of compact sets.
An alternative perspective of this “largeness” pertains to a certain removability of sets. A compact set is considered to be removable if every bounded holomorphic function on the complement can be extended to . For instance, the analytic capacity of any singleton (any singular point) or set of discrete points is 0, as evidenced by Proposition 6.3.1; and moreover, any singleton or compact set of discrete points is a removable set. In a heuristic sense, analytic capacity measures the irremovability of a set, and larger sets tend to be “less removable.”
A compact set is a continuum if it is connected, is connected, and if it is not a singleton ( contains at least 2 distinct points).
Proposition 6.3.2.
Let be a continuum. Then where is a biholomorphism satisfying (i. e. the maximal in the supremum of the definition of analytic capacity is attained when is biholomorphic).
Proof.
Let be the biholomorphism, be holomorphic (not necessarily surjective) mapping to . Since and maps to , the Schwarz Lemma (Lemma 3.5.1) implies that
for all . Thus, , and
Proposition 6.3.3.
The analytic capacity of any closed disk is the radius.
Proof.
Since is a continuum, a biholomorphism such that . One such biholomorphism is given by
Hence, Proposition 6.3.2, gives that .
Proposition 6.3.4.
If is a continuum, then
Proof.
Assume is a biholomorphism mapping to . The lower bound follows directly from Proposition 6.2.1. Let be arbitrary, then for any , we obtain , implying that . By Proposition 6.3.3, we have , and Theorem 6.3.1 consequently gives the upper bound of
We outline the precise connections to rational approximation:
Theorem 6.3.2.
Let be compact such that such that , ,
Then holomorphic on can be uniformly and rationally approximated on with poles in .
Corollary 6.3.1.
Let be compact. If the connected components of give the uniform existence of some such that , , then holomorphic on can be uniformly and rationally approximated on with poles in .
Notice here that no restrictions are imposed on the finiteness of the number of connected components of the complement. The general conclusion given for Mergelyan’s Theorem is not true for more general compact sets.
The counterexample we now provide due to [1], we provide the construction of the compact set .
Example 6.3.1.
There exists a compact set and , such that is holomorphic on and cannot be rationally approximated on .
Proof.
Let be a countably dense set of points in (use a bijection and Cantor’s pairing function to get a surjection ).
Fix . Let and , . For each , let be the first to be the first point in the dense sequence such that
Then choose such that lies in and does not intersect any previous for (possible by the fact that each does not lie on the boundary of the previous disks) and so that
under the inductive hypothesis that . Apply another bound, so that
under the additional assumption that . Repeat this process inductively for all . Define
which is compact. For any point , no disk centered at exists such that is contained in , since a subsequence of accumulating to in is removed from . Hence, . Hence, any is holomorphic on the interior.
(This general construction of is known as the Swiss cheese set)
We now show that cannot be uniformly rationally approximated on . By explicit calculation or Green’s Theorem (Theorem 3.1.2), we have
and by similar reasoning
For any rational with poles off ,
by Theorem 3.1.5. The summation’s convergence follows from being termwise absolutely bounded by , which converges by construction. Now, if on , then
which is impossible.