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 (i.e. the boundary of ) can be approximated uniformly on by (real-valued) harmonic polynomials in the real variables and .
Generalizations The Walsh–Lebesgue theorem has been generalized to Riemann surfaces and to (real coordinate space).
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.
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