Walsh–Lebesgue theorem

The Walsh–Lebesgue theorem is a famous result from harmonic analysis proved by the American mathematician Joseph L. Walsh in 1929, using results proved by Lebesgue in 1907. The theorem states the following:

Let $K$ be a compact subset of the Euclidean plane $ℝ^{2}$ such the relative complement of $$K$$ with respect to $ℝ^{2}$ is connected. Then, every real-valued continuous function on $$\partial{K}$$ (i.e. the boundary of $K$) can be approximated uniformly on $$\partial{K}$$ by (real-valued) harmonic polynomials in the real variables $x$ and $y$.

Generalizations
The Walsh–Lebesgue theorem has been generalized to Riemann surfaces and to $ℝ^{n}$.

"This Walsh-Lebesgue theorem has also served as a catalyst for entire chapters in the theory of function algebras such as the theory of Dirichlet algebras and logmodular algebras."

In 1974 Anthony G. O'Farrell gave a generalization of the Walsh–Lebesgue theorem by means of the 1964 Browder–Wermer theorem with related techniques.