View source for Lebesgue constants

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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|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|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|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|Lebesgue function]].

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  A. Zygmund,   "Trigonometric series" , '''1''' , Cambridge Univ. Press  (1988)</TD></TR>
<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>

====Comments====

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''

====Comments====
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. [[#References|[a1]]]); historical notes are also included there.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> T.A. Kilgore, "A characterization of the Lagrange interpolation projection with minimal Tchebycheff norm"  ''J. Approx. Theory'' , '''24'''  (1978)  pp. 273–288</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>
<TR><TD valign="top">[a3]</TD> <TD valign="top"> Steven R. Finch, ''Mathematical Constants'', Cambridge University Press (2003) {{ISBN|0-521-81805-2}}.  Sect. 4.2</TD></TR>

</table>