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.
|||S. Kaczmarz, H. Steinhaus, "Theorie der Orthogonalreihen" , Chelsea, reprint (1951)|
Lebesgue function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lebesgue_function&oldid=13866