# Lebesgue function

From Encyclopedia of Mathematics

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

where

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.

#### References

[1] | S. Kaczmarz, H. Steinhaus, "Theorie der Orthogonalreihen" , Chelsea, reprint (1951) |

**How to Cite This Entry:**

Lebesgue function. B.S. Kashin (originator),

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Lebesgue_function&oldid=13866

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098