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 [2]. 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 [1].

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 *H*_{2} the
vector space of analytic functions *f*(*z*) in *D* that have a
finite norm

Let

about any point

over circles

Our version of Runge's theorem asserts that the class *P* of
polynomials is dense in *H*_{2}. If this were false then by the
projection theorem there would exist a nontrivial function *f*in *H*_{2} orthogonal to all the elements of *P*. From the expansion
of 1/(*z*-*t*) in a geometric series of powers of *z* we
conclude that

when

Because *f* is analytic, a direct calculation shows that

for every

Let *E* be a circle inside *D* touching *C* at the point *t*_{0},
and suppose *t* and *t*^{*} are inverse points with respect to *E*situated near *t*_{0}. Relocating the origin at the center
of *E*, we obtain the relation

for

On the other hand, according to the Schwarz inequality

the two integrals

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 *z*_{0} 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/(

for

there, which suffices to prove the desired continuity of

- 1.
P. Garabedian,
*Partial Differential Equations,*Amer. Math. Soc., 1998. - 2.
Z. Nehari,
*Conformal Mapping,*Dover, New York, 1975.