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 ℝ² such the relative complement of ${\displaystyle K}$ with respect to ℝ² 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 ℝⁿ.

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]

References

1. Walsh, J. L. (1928). "Über die Entwicklung einer harmonischen Funktion nach harmonischen Polynomen". J. Reine Angew. Math. 159: 197–209.
2. Walsh, J. L. (1929). "The approximation of harmonic functions by harmonic polynomials and by harmonic rational functions". Bull. Amer. Math. Soc. 35 (2): 499–544. doi:10.1090/S0002-9947-1929-1501495-4.
3. Lebesgue, H. (1907). "Sur le probléme de Dirichlet". Rendiconti del Circolo Matematico di Palermo. 24 (1): 371–402. doi:10.1007/BF03015070. S2CID 120228956.
4. Gamelin, Theodore W. (1984). "3.3 Theorem (Walsh-Lebesgue Theorem)". Uniform Algebras. American Mathematical Society. pp. 36–37. ISBN 9780821840498.
5. Bagby, T.; Gauthier, P. M. (1992). "Uniform approximation by global harmonic functions". Approximations by solutions of partial differential equations. Dordrecht: Springer. pp. 15–26 (p. 20). ISBN 9789401124362.
6. Walsh, J. L. (2000). Rivlin, Theodore J.; Saff, Edward B. (eds.). Joseph L. Walsh. Selected papers. Springer. pp. 249–250. ISBN 978-0-387-98782-8.
7. Browder, A.; Wermer, J. (August 1964). "A method for constructing Dirichlet algebras". Proceedings of the American Mathematical Society. 15 (4): 546–552. doi:10.1090/s0002-9939-1964-0165385-0. JSTOR 2034745.
8. O'Farrell, A. G (2012). "A Generalised Walsh-Lebesgue Theorem" (PDF). Proceedings of the Royal Society of Edinburgh, Section A. 73: 231–234. doi:10.1017/S0308210500016395.
9. O'Farrell, A. G. (1981). "Five Generalisations of the Weierstrass Approximation Theorem" (PDF). Proceedings of the Royal Irish Academy, Section A. 81 (1): 65–69.
10. O'Farrell, A. G. (1980). "Theorems of Walsh-Lebesgue Type" (PDF). In D. A. Brannan; J. Clunie (eds.). Aspects of Contemporary Complex Analysis. Academic Press. pp. 461–467.