Lebesgue function

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A function

where is a given system of functions, orthonormal with respect to the Lebesgue measure on the interval , . Lebesgue functions are defined similarly in the case when an orthonormal system is specified on an arbitrary measure space. One has


is the -th partial sum of the Fourier series of with respect to . In the case when is the trigonometric system, the Lebesgue functions are constant and reduce to the Lebesgue constants. They were introduced by H. Lebesgue.


[1] S. Kaczmarz, H. Steinhaus, "Theorie der Orthogonalreihen" , Chelsea, reprint (1951)
How to Cite This Entry:
Lebesgue function. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by B.S. Kashin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article