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 ℝ2 such the relative complement of formula_1 with respect to ℝ2 is connected. Then, every real-valued continuous function on formula_2 ("i.e." the boundary of "K") can be approximated uniformly on formula_2 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 ℝn.
 ;templatestyles src="Template:Blockquote/styles.css" / ;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.