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.
[1][2][3] The theorem states the following: Let K be a compact subset of the Euclidean plane ℝ2 such the relative complement of
by (real-valued) harmonic polynomials in the real variables x and y.
[4] The Walsh–Lebesgue theorem has been generalized to Riemann surfaces[5] and to ℝn.
[6]In 1974 Anthony G. O'Farrell gave a generalization of the Walsh–Lebesgue theorem by means of the 1964 Browder–Wermer theorem[7] with related techniques.