Approximating harmonic functions on Rn with one function of a single complex variable

✍ Scribed by R. J. Whitley; T. V. Hromadka II

• Publisher: John Wiley and Sons
• Year: 2005
• Tongue: English
• Weight: 114 KB
• Volume: 21
• Category: Article
• ISSN: 0749-159X
• DOI: 10.1002/num.20067

✦ Synopsis

Let ⍀ be a bounded open set in R n , n Ͼ 2, with R n Ϫ ⍀ a connected set that is not thin at each point of Ѩ⍀. Then any solution to a Dirichlet problem for given continuous boundary data on Ѩ⍀ can be approximated in a simple way by a sum that involves one function f (z) of a single complex variable z; any analytic function f(z) not a polynomial can be used. One consequence of this approximation property is that any harmonic polynomial can be written (exactly, not approximately) as a finite sum involving polynomials in one complex variable. These results reveal an unexpected simplicity in the structure of harmonic functions on R n . It is a common and simple observation that harmonic analysis is more difficult in three or more dimensions than in two dimensions because you do not have the direct use of the theory of analytic functions of a single complex variable; the results here show that this obvious observation is not correct.

For applications, these approximating sums provide a large collection of functions that can be fit to given boundary data and used for the numerical solution of Dirichlet problems in R n .

The harmonic functions considered will be real-valued functions of a variable in R n , typically x ϭ (x 1 , x 2 , . . . , x n ), with n Ͼ 2.

The Walsh-Lebesgue Theorem states that if K is a compact subset of R 2 , with R 2 Ϫ K connected, then every continuous real-valued function on ѨK can be uniformly approximated by functions of the form Re P(z), P(z) a polynomial in the complex variable z. Theorem 1 generalizes this result to R n , with a proof that is a modification of the proof given in [1, Corollary 6.3.4, p 173] for R 2 .

Theorem 1. Let ⍀ be a bounded domain in R n , n Ͼ 2, with R n Ϫ ⍀ connected. Suppose that R n Ϫ ⍀ is not thin at any point of Ѩ⍀.

(1)