Lebesgue constants
The quantities
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where
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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
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for all functions such that
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Here is the
-th partial sum of the trigonometric Fourier series of the
-periodic function
. The following asymptotic formula is valid:
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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
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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
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where
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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
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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
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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
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then
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The Lebesgue constants of an arbitrary interval
are connected with the analogous constants
for the interval
by the relation
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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 |
Lebesgue constants. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lebesgue_constants&oldid=47600