# Lebesgue constants

The quantities

$$L _ {n} = \frac{1} \pi \int\limits _ {- \pi } ^ \pi | D _ {n} ( t) | dt ,$$

where

$$D _ {n} ( t) = \frac{\sin \left ( \frac{2n + 1 }{2} t \right ) }{2 \sin ( t/2 ) }$$

is the Dirichlet kernel. The Lebesgue constants $L _ {n}$ for each $n$ equal:

1) the maximum value of $| S _ {n} ( f , x ) |$ for all $x$ and all continuous functions $f$ such that $| f ( t) | \leq 1$ for almost-all $t$;

2) the least upper bound of $| S _ {n} ( f , x ) |$ for all $x$ and all continuous functions $f$ such that $| f ( t) | \leq 1$;

3) the least upper bound of the integrals

$$\int\limits _ { 0 } ^ { {2 } \pi } | S _ {n} ( f , x ) | dx$$

for all functions $f$ such that

$$\int\limits _ { 0 } ^ { {2 } \pi } | f ( t) | dt \leq 1 .$$

Here $S _ {n} ( f , x )$ is the $n$- th partial sum of the trigonometric Fourier series of the $2 \pi$- periodic function $f$. The following asymptotic formula is valid:

$$L _ {n} = \frac{4}{\pi ^ {2} } \mathop{\rm ln} n + O ( 1) ,\ n \rightarrow \infty .$$

In particular, $L _ {n} \rightarrow \infty$ as $n \rightarrow \infty$; this is connected with the divergence of the trigonometric Fourier series of certain continuous functions. In a wider sense the Lebesgue constants are defined for other orthonormal systems (cf. Orthogonal system) as the quantities

$$L _ {n} = \mathop{\rm esssup} _ {x \in ( a , b ) } \int\limits _ { a } ^ { b } | D _ {n} ( x , t ) | dt ,$$

where $D _ {n} ( x , t )$ is the Dirichlet kernel for the given orthonormal system of functions on $( a , b )$; they play an important role in questions of convergence of Fourier series in these systems. The Lebesgue constants were introduced by H. Lebesgue (1909). See also Lebesgue function.

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How to Cite This Entry:
Lebesgue constants. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lebesgue_constants&oldid=47600
This article was adapted from an original article by K.I. Oskolkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article