Lebesgue constants

The quantities

where

is the Dirichlet kernel. The Lebesgue constants for each equal:

1) the maximum value of for all and all continuous functions such that for almost-all ;

2) the least upper bound of for all and all continuous functions such that ;

3) the least upper bound of the integrals

for all functions such that

Here is the -th partial sum of the trigonometric Fourier series of the -periodic function . The following asymptotic formula is valid:

In particular, as ; 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

where is the Dirichlet kernel for the given orthonormal system of functions on ; 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.

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

Comments

The Lebesgue constants of an interpolation process are the numbers

where

and are pairwise distinct interpolation points lying in some interval .

Let and be, respectively, the space of continuous functions on and the space of algebraic polynomials of degree at most , considered on the same interval, with the uniform metric, and let be the interpolation polynomial of degree that takes the same values at the points , , as . If denotes the operator that associates with , i.e. , then , where the left-hand side is the operator norm in the space of bounded linear operators and

where is the best approximation of by algebraic polynomials of degree at most .

For any choice of the interpolation points in , one has . For equidistant points a constant exists such that . In case of the interval , for points coinciding with the zeros of the -th Chebyshev polynomial, the Lebesgue constants have minimum order of growth, namely

If is times differentiable on , is a given set of numbers ( "approximations of the values fxk" ), is the interpolation polynomial of degree that takes the values at the points , , and

then

The Lebesgue constants of an arbitrary interval are connected with the analogous constants for the interval by the relation

in particular, .

L.D. Kudryavtsev

Comments

The problem to determine "optimal nodes" , i.e., for a fixed positive integer , to determine such that is minimal, has been given much attention. S.N. Bernstein [S.N. Bernshtein] (1931) conjectured that is minimal when "equi-oscillates" . Bernstein's conjecture was proved by T.A. Kilgore (cf. [a1]); historical notes are also included there.

References

[a1]	T.A. Kilgore, "A characterization of the Lagrange interpolation projection with minimal Tchebycheff norm" J. Approx. Theory , 24 (1978) pp. 273–288
[a2]	T.J. Rivlin, "An introduction to the approximation of functions" , Blaisdell (1969) pp. Sect. 4.2
[a3]	Steven R. Finch, Mathematical Constants, Cambridge University Press (2003) ISBN 0-521-81805-2. Sect. 4.2