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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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

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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.

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