Walsh–Lebesgue theorem explained

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 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 .^[4]

Generalizations

The Walsh–Lebesgue theorem has been generalized to Riemann surfaces^[5] and to .

In 1974 Anthony G. O'Farrell gave a generalization of the Walsh–Lebesgue theorem by means of the 1964 Browder–Wermer theorem^[6] with related techniques.^{[7] [8] [9]}

References

1. Walsh, J. L.. Über die Entwicklung einer harmonischen Funktion nach harmonischen Polynomen. J. Reine Angew. Math.. 1928. 159. 197–209.
2. Walsh, J. L.. The approximation of harmonic functions by harmonic polynomials and by harmonic rational functions. Bull. Amer. Math. Soc.. 1929. 35. 2. 499–544. 10.1090/S0002-9947-1929-1501495-4. free.
3. Lebesgue, H.. Sur le probléme de Dirichlet. Rendiconti del Circolo Matematico di Palermo. 24. 1. 1907. 371–402. 10.1007/BF03015070. 120228956.
4. Book: Gamelin, Theodore W.. Theodore Gamelin. 3.3 Theorem (Walsh-Lebesgue Theorem). Uniform Algebras. 1984. 36–37. American Mathematical Society. 9780821840498. https://books.google.com/books?id=2-K2A7cdORoC&pg=PA36.
5. Book: Bagby, T.. Gauthier, P. M.. Uniform approximation by global harmonic functions. Approximations by solutions of partial differential equations. Springer. Dordrecht. 1992. https://books.google.com/books?id=vzrsCAAAQBAJ&pg=PA20. 15–26 (p. 20). 9789401124362.
6. Browder, A.. Andrew Browder. Wermer, J.. John Wermer. A method for constructing Dirichlet algebras. Proceedings of the American Mathematical Society. 15. 4. August 1964. 546–552. 10.1090/s0002-9939-1964-0165385-0. 2034745. free.
7. A Generalised Walsh-Lebesgue Theorem. 10.1017/S0308210500016395. Proceedings of the Royal Society of Edinburgh, Section A. 73. 231–234. 2012. O'Farrell. A. G.
8. O'Farrell, A. G.. Five Generalisations of the Weierstrass Approximation Theorem. Proceedings of the Royal Irish Academy, Section A . 81. 1. 1981. 65–69.
9. Book: O'Farrell, A. G. . Theorems of Walsh-Lebesgue Type . Aspects of Contemporary Complex Analysis . D. A. Brannan . J. Clunie . 1980. 461–467. Academic Press. http://archive.maths.nuim.ie/staff/aof/preprint/1980towlt.pdf.