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(→‎References: Finch (2003))
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====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.W. Cheney,  "Introduction to approximation theory" , McGraw-Hill  (1966)  pp. Chapts. 4&amp;6</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T.J. Rivlin,  "An introduction to the approximation of functions" , Blaisdell  (1969)  pp. Sect. 4.2</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.W. Cheney,  "Introduction to approximation theory" , McGraw-Hill  (1966)  pp. Chapts. 4&amp;6</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T.J. Rivlin,  "An introduction to the approximation of functions" , Blaisdell  (1969)  pp. Sect. 4.2</TD></TR></table>
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====Comments====
  
 
The Lebesgue constants of an interpolation process are the numbers
 
The Lebesgue constants of an interpolation process are the numbers
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and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780029.png" /> are pairwise distinct interpolation points lying in some interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780030.png" />.
 
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780029.png" /> are pairwise distinct interpolation points lying in some interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780030.png" />.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780032.png" /> be, respectively, the space of continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780033.png" /> and the space of algebraic polynomials of degree at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780034.png" />, considered on the same interval, with the uniform metric, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780035.png" /> be the interpolation polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780036.png" /> that takes the same values at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780038.png" />, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780039.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780040.png" /> denotes the operator that associates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780041.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780042.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780043.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780044.png" />, where the left-hand side is the operator norm in the space of bounded linear operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780045.png" /> and
+
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780032.png" /> be, respectively, the space of continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780033.png" /> and the space of algebraic polynomials of degree at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780034.png" />, considered on the same interval, with the uniform metric, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780035.png" /> be the interpolation polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780036.png" /> that takes the same values at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780038.png" />, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780039.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780040.png" /> denotes the operator that associates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780041.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780042.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780043.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780044.png" />, where the left-hand side is the [[operator norm]] in the space of bounded linear operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780045.png" /> and
  
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780046.png" /></td> </tr></table>
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780046.png" /></td> </tr></table>

Revision as of 17:10, 29 October 2017

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)


Comments

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
How to Cite This Entry:
Lebesgue constants. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lebesgue_constants&oldid=42218
This article was adapted from an original article by K.I. Oskolkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article