The Walsh–Lebesgue theorem is a famous result from
harmonic analysis proved by the American mathematician
Joseph L. Walsh
__NOTOC__
Joseph Leonard Walsh (September 21, 1895 – December 6, 1973) was an American mathematician who worked mainly in the field of analysis. The Walsh function and the Walsh–Hadamard code are named after him. The Grace–Walsh–Szegő ...
in 1929, using results proved by
Lebesgue
Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
in 1907. The theorem states the following:
Let ' be a
compact subset
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i. ...
of the
Euclidean plane such the
relative complement
In set theory, the complement of a set , often denoted by (or ), is the set of elements not in .
When all sets in the universe, i.e. all sets under consideration, are considered to be members of a given set , the absolute complement of is th ...
of
with respect to is
connected
Connected may refer to:
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* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
. Then, every real-valued
continuous function on
(''i.e.'' the
boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
* ''Boundaries'' (2016 film), a 2016 Canadian film
* ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film
*Boundary (cricket), the edge of the pla ...
of ') can be
approximated uniformly on
by (real-valued)
harmonic polynomial In mathematics, in abstract algebra, a multivariate polynomial over a field such that the Laplacian of is zero is termed a harmonic polynomial.
The harmonic polynomials form a vector subspace of the vector space of polynomials over the field. In ...
s in the real variables and .
Generalizations
The Walsh–Lebesgue theorem has been generalized to
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
s and to .
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.
References
{{DEFAULTSORT:Walsh-Lebesgue theorem
Theorems in harmonic analysis
Theorems in approximation theory