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.^[1][2][3] The theorem states the following:

Let K be a compact subset of the Euclidean plane ${\displaystyle \mathbb {R} ^{2}}$ such the relative complement of ${\displaystyle K}$ with respect to ${\displaystyle \mathbb {R} ^{2}}$ is connected. Then, every real-valued continuous function on ${\displaystyle \partial {K}}$ (i.e. the boundary of K) can be approximated uniformly on ${\displaystyle \partial {K}}$ by (real-valued) harmonic polynomials in the real variables x and y.^[4]

Generalizations

The Walsh–Lebesgue theorem has been generalized to Riemann surfaces^[5] and to ${\displaystyle \mathbb {R} ^{n}}$ (real coordinate space).

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.^[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.^[8][9][10]