Lebesgue function

A function

$$L _ {n} ^ \Phi ( t) = \int\limits _ { a } ^ { b } \left | \sum _ { k=1 } ^ { n } \phi _ {k} ( x) \phi _ {k} ( t) \right | d x ,\ t \in [ a , b ] ,$$

where $\Phi = \{ \phi _ {k} \} _ {k=1} ^ \infty$ is a given system of functions, orthonormal with respect to the Lebesgue measure on the interval $[ a , b ]$, $n = 1 , 2 , . . .$. Lebesgue functions are defined similarly in the case when an orthonormal system is specified on an arbitrary measure space. One has

$$L _ {n} ^ \Phi ( t) = \ \sup _ {f : \| f \| _ {C [ a , b ] } \leq 1 } | S _ {n} ( f ) | ,\ \ t \in [ a , b ] ,$$

where

$$S _ {n} ( f ) ( t) = \sum _ { k=1 } ^ { n } c _ {k} ( f ) \phi _ {k} ( t)$$

is the $n$-th partial sum of the Fourier series of $f$ with respect to $\Phi$. In the case when $\Phi$ 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)