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 be a compact subset of the Euclidean plane such the relative complement of with respect to is connected. Then, every real-valued continuous function on can be approximated uniformly on by harmonic polynomials in the real variables and.

Generalizations

The Walsh–Lebesgue theorem has been generalized to Riemann surfaces and to .
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.