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.

Contents

References

 [1] A. Zygmund, "Trigonometric series" , 1 , Cambridge Univ. Press (1988)

References

 [a1] E.W. Cheney, "Introduction to approximation theory" , McGraw-Hill (1966) pp. Chapts. 4&6 [a2] T.J. Rivlin, "An introduction to the approximation of functions" , Blaisdell (1969) pp. Sect. 4.2

The Lebesgue constants of an interpolation process are the numbers

$$\lambda _ {n} = \max _ {a \leq x \leq b } \sum _ { k= } 0 ^ { n } | l _ {nk} ( x) | ,\ n = 1 , 2 \dots$$

where

$$l _ {nk} ( x) = \prod _ {j \neq k } \frac{x - x _ {j} }{x _ {k} - x _ {j} }$$

and $x _ {0} \dots x _ {n}$ are pairwise distinct interpolation points lying in some interval $[ a , b ]$.

Let $C [ a , b ]$ and ${\mathcal P} _ {n} [ a , b ]$ be, respectively, the space of continuous functions on $[ a , b ]$ and the space of algebraic polynomials of degree at most $n$, considered on the same interval, with the uniform metric, and let $P _ {n} ( x , f )$ be the interpolation polynomial of degree $\leq n$ that takes the same values at the points $x _ {k}$, $k = 0 \dots n$, as $f$. If $P _ {n}$ denotes the operator that associates $P _ {n} ( x , f )$ with $f ( x)$, i.e. $P _ {n} : C [ a , b ] \rightarrow {\mathcal P} _ {n} [ a , b ]$, then $\| P _ {n} \| = \lambda _ {n}$, where the left-hand side is the operator norm in the space of bounded linear operators ${\mathcal L} ( C [ a , b ] , P _ {n} [ a , b ] )$ and

$$\| f ( x) - P _ {n} ( x , f ) \| _ {C [ a , b ] } \leq ( 1 + \lambda _ {n} ) E _ {n} ( f ) ,$$

where $E _ {n} ( f )$ is the best approximation of $f$ by algebraic polynomials of degree at most $n$.

For any choice of the interpolation points in $[ a , b ]$, one has $\lim\limits _ {n \rightarrow \infty } \lambda _ {n} = + \infty$. For equidistant points a constant $c > 0$ exists such that $\lambda _ {n} \geq c 2 ^ {n} n ^ {- 3/2 }$. In case of the interval $[ - 1 , 1 ]$, for points coinciding with the zeros of the $n$- th Chebyshev polynomial, the Lebesgue constants have minimum order of growth, namely

$$\lambda _ {n} \approx \mathop{\rm ln} n .$$

If $f$ is $m$ times differentiable on $[ a , b ]$, $Y = \{ y _ {k} \} _ {k=} 0 ^ {n}$ is a given set of numbers ( "approximations of the values fxk" ), $P _ {n} ( x , Y )$ is the interpolation polynomial of degree $\leq n$ that takes the values $y _ {k}$ at the points $x _ {k}$, $k = 0 \dots n$, and

$$\lambda _ {nm} = \max _ {a \leq x \leq b } \sum _ { k= } 1 ^ { n } | l _ {nk} ^ {(} m) ( x) | ,\ n = 0 , 1 \dots$$

then

$$\| f ^ { ( m) } ( x) - P _ {n} ^ {(} m) ( x , Y ) \| _ {C [ a , b ] } \leq$$

$$\leq \ \| f ^ { ( m) } ( x) - P _ {n} ^ {(} m) ( x , f ) \| _ {C [ a , b ] } +$$

$$+ \lambda _ {nm} \max _ {k = 0 \dots n } | f ( x _ {k} ) - y _ {k} | .$$

The Lebesgue constants $\lambda _ {nm}$ of an arbitrary interval $[ a , b ]$ are connected with the analogous constants $\Lambda _ {nm}$ for the interval $[ - 1 , 1 ]$ by the relation

$$\Lambda _ {nm} = \left ( b- \frac{a}{2} \right ) ^ {m} \lambda _ {nm} ;$$

in particular, $\lambda _ {n} = \Lambda _ {n0}$.

L.D. Kudryavtsev

The problem to determine "optimal nodes" , i.e., for $n$ a fixed positive integer $\geq 2$, to determine $x _ {0} \dots x _ {n}$ such that $\lambda _ {n}$ is minimal, has been given much attention. S.N. Bernstein [S.N. Bernshtein] (1931) conjectured that $\lambda _ {n}$ is minimal when $\sum _ {k=} 0 ^ {n} | l _ {n k } ( x) |$" equi-oscillates" . Bernstein's conjecture was proved by T.A. Kilgore (cf. [a1]); historical notes are also included there.