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

Generalizations

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

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.