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.{{cite journal|author=Walsh, J. L.|title=Über die Entwicklung einer harmonischen Funktion nach harmonischen Polynomen|journal=J. Reine Angew. Math.|year=1928|volume=159|pages=197–209|url=http://eudml.org/doc/149665}}{{cite journal|author=Walsh, J. L.|title=The approximation of harmonic functions by harmonic polynomials and by harmonic rational functions|journal=Bull. Amer. Math. Soc.|year=1929|volume=35|issue=2|pages=499–544|doi=10.1090/S0002-9947-1929-1501495-4|doi-access=free}}{{cite journal|author=Lebesgue, H.|title=Sur le probléme de Dirichlet|journal=Rendiconti del Circolo Matematico di Palermo|volume=24|issue=1|year=1907|pages=371–402|doi=10.1007/BF03015070|s2cid=120228956|url=https://zenodo.org/record/2189809}} The theorem states the following:
Let {{math|K}} be a compact subset of the Euclidean plane {{math|ℝ2}} such the relative complement of with respect to {{math|ℝ2}} is connected. Then, every real-valued continuous function on (i.e. the boundary of {{math|K}}) can be approximated uniformly on by (real-valued) harmonic polynomials in the real variables {{math|x}} and {{math|y}}.{{cite book|author=Gamelin, Theodore W.|author-link=Theodore Gamelin|chapter=3.3 Theorem (Walsh-Lebesgue Theorem)|title=Uniform Algebras|year=1984|pages=36–37|publisher=American Mathematical Society|isbn=9780821840498|chapter-url=https://books.google.com/books?id=2-K2A7cdORoC&pg=PA36}}
Generalizations
The Walsh–Lebesgue theorem has been generalized to Riemann surfaces{{cite book|author=Bagby, T.|author2=Gauthier, P. M.|chapter=Uniform approximation by global harmonic functions|title=Approximations by solutions of partial differential equations|publisher=Springer|location=Dordrecht|year=1992|chapter-url=https://books.google.com/books?id=vzrsCAAAQBAJ&pg=PA20|pages=15–26 (p. 20)|isbn=9789401124362}} and to {{math|ℝn}}.
{{blockquote|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.{{cite book|author=Walsh, J. L.|editor=Rivlin, Theodore J.|editor-link=Theodore J. Rivlin|editor2=Saff, Edward B.|editor-link2=Edward B. Saff|title=Joseph L. Walsh. Selected papers|year=2000|publisher=Springer|pages=249–250|url=https://books.google.com/books?id=Zm6ahqIyF5QC&pg=PA249|isbn=978-0-387-98782-8}}}}
In 1974 Anthony G. O'Farrell gave a generalization of the Walsh–Lebesgue theorem by means of the 1964 Browder–Wermer theorem{{cite journal|author=Browder, A.|author-link=Andrew Browder|author2=Wermer, J.|authorlink2=John Wermer|title=A method for constructing Dirichlet algebras|journal=Proceedings of the American Mathematical Society|volume=15|issue=4|date=August 1964|pages=546–552|doi=10.1090/s0002-9939-1964-0165385-0|jstor=2034745|doi-access=free}} with related techniques.{{cite journal|url=http://archive.maths.nuim.ie/staff/aofarrell/preprint/1974agwlt.pdf |doi=10.1017/S0308210500016395|title=A Generalised Walsh-Lebesgue Theorem|journal=Proceedings of the Royal Society of Edinburgh, Section A|volume=73|pages=231–234|year=2012|last1=O'Farrell|first1=A. G}}{{cite journal|author=O'Farrell, A. G.|title=Five Generalisations of the Weierstrass Approximation Theorem|journal=Proceedings of the Royal Irish Academy, Section A |volume=81|issue=1|year=1981|pages=65–69|url=http://archive.maths.nuim.ie/staff/aof/preprint/19815gotwat.pdf}}{{cite book|chapter=Theorems of Walsh-Lebesgue Type |first=A. G. |last=O'Farrell |title=Aspects of Contemporary Complex Analysis |editor1=D. A. Brannan |editor2=J. Clunie |year=1980|pages=461–467|publisher=Academic Press|chapter-url=http://archive.maths.nuim.ie/staff/aof/preprint/1980towlt.pdf}}