RUNGE'S THEOREM AND THE RIEMANN MAPPING THEOREM
The standard formulation of Runge's theorem asserts that an analytic
function can be approximated by polynomials in the maximum norm
over any simply-connected bounded domain. The usual proof of a strong
form of the result appeals to the Riemann mapping theorem . In
this note we shall present a least squares version based on the
elementary theory of Hilbert space. Instead of using the mapping
theorem, our approach provides an alternate way of proving it .
Let D be a domain of the complex plane bounded by a closed
curve C with the smoothness property that each of its points
can be touched by circles E of a fixed radius R located
in either the interior or the exterior of D. Denote by H2 the
vector space of analytic functions f(z) in D that have a
Let f(z) be expanded in a Taylor series
about any point z0 of D. From the estimate
over circles E inside D one sees that every Cauchy
sequence in H2 converges uniformly in each closed subdomain
of D. It follows that H2 is a complete Hilbert space.
Our version of Runge's theorem asserts that the class P of
polynomials is dense in H2. If this were false then by the
projection theorem there would exist a nontrivial function fin H2 orthogonal to all the elements of P. From the expansion
of 1/(z-t) in a geometric series of powers of z we
when t is sufficiently large. By the principle of analytic
continuation this identity must remain true for all
choices of t in the exterior of D.
Because f is analytic, a direct calculation shows that
for every t in the interior of D. It follows that I is
a harmonic function throughout the domain D. If we can
establish that the integral I is continuous across the
curve C, it has to vanish there and becomes identically zero
in D. This would complete our proof of Runge's theorem by
contradicting the assumption that f is nontrivial.
Let E be a circle inside D touching C at the point t0,
and suppose t and t* are inverse points with respect to Esituated near t0. Relocating the origin at the center
of E, we obtain the relation
for z on the boundary of E. From an application
of Green's theorem we conclude that
On the other hand, according to the Schwarz inequality
the two integrals I(t) and I(t*) differ
over the complementary
region D-E by an amount approaching zero as t and t*approach t0. Combined with the fact that I vanishes
outside C, this is enough to show that it has zero boundary
values on C. That was precisely what we needed for the proof
of the Hilbert space formulation of Runge's theorem.
The method we have described enables one to prove the Riemann
mapping theorem in a way that makes it easy to establish continuity
of both the real and the imaginary parts of the map function
at the boundary. To see this
let z0 be a point inside the region D, and apply the
Riesz representation theorem to conclude that the linear functional
defined by the singular kernel 1/(z-z0) can be expressed as a scalar
product in H with some regular element g.
p'=1/(z-z0) - g is orthogonal to every element of H, the
analysis given before proves that it is the logarithmic derivative
of the map function. The result hinges on the formula
for p itself, where the last two integrals on the right are,
respectively, an analytic function of t and an analytic function
in the tangent circle E. From the principle of analytic
continuation we conclude that
there, which suffices to prove the desired continuity of p at
the boundary C because of the estimate of this kind of integral
presented above. The simplicity of the proof depends, of course, on the
strong assumption that the curve C has finite curvature.
P. Garabedian, Partial Differential Equations, Amer. Math.
Z. Nehari, Conformal Mapping, Dover, New York, 1975.