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The quantities
 
The quantities
  
<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/l0578001.png" /></td> </tr></table>
+
$$
 +
L _ {n}  =
 +
\frac{1} \pi
 +
\int\limits _ {- \pi } ^  \pi  | D _ {n} ( t) |  dt ,
 +
$$
  
 
where
 
where
  
<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/l0578002.png" /></td> </tr></table>
+
$$
 +
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l0578003.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l0578004.png" /> equal:
+
is the [[Dirichlet kernel|Dirichlet kernel]]. The Lebesgue constants $  L _ {n} $
 +
for each $  n $
 +
equal:
  
1) the maximum value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l0578005.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l0578006.png" /> and all continuous functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l0578007.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l0578008.png" /> for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l0578009.png" />;
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780010.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780011.png" /> and all continuous functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780012.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780013.png" />;
+
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
 
3) the least upper bound of the integrals
  
<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/l05780014.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { 0 } ^ { {2 }  \pi } | S _ {n} ( f , x ) |  dx
 +
$$
  
for all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780015.png" /> such that
+
for all functions $  f $
 +
such that
  
<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/l05780016.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { 0 } ^ { {2 }  \pi } | f ( t) |  dt  \leq  1 .
 +
$$
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780017.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780018.png" />-th partial sum of the trigonometric [[Fourier series|Fourier series]] of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780019.png" />-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780020.png" />. The following asymptotic formula is valid:
+
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:
  
<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/l05780021.png" /></td> </tr></table>
+
$$
 +
L _ {n}  =
 +
\frac{4}{\pi  ^ {2} }
 +
  \mathop{\rm ln}  n + O ( 1) ,\  n \rightarrow \infty .
 +
$$
  
In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780022.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780023.png" />; 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
+
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
  
<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/l05780024.png" /></td> </tr></table>
+
$$
 +
L _ {n}  =   \mathop{\rm esssup} _ {x \in ( a , b ) }  \int\limits _ { a } ^ { b }  | D _ {n} ( x , t ) |  dt ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780025.png" /> is the Dirichlet kernel for the given orthonormal system of functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780026.png" />; 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]].
+
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====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1''' , Cambridge Univ. Press  (1988)</TD></TR></table>
+
<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====
 
 
 
 
 
====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>
 
  
 
====Comments====
 
====Comments====
Line 46: Line 91:
 
The Lebesgue constants of an interpolation process are the numbers
 
The Lebesgue constants of an interpolation process are the numbers
  
<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/l05780027.png" /></td> </tr></table>
+
$$
 +
\lambda _ {n}  = \max _ {a \leq  x \leq  b }  \sum _ { k= 0} ^ { n }  |
 +
l _ {nk} ( x) | ,\  n = 1 , 2 \dots
 +
$$
  
 
where
 
where
  
<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/l05780028.png" /></td> </tr></table>
+
$$
 +
l _ {nk} ( x)  = \prod _ {j \neq k }
 +
\frac{x - x _ {j} }{x _ {k} - x _ {j} }
  
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
+
and $  x _ {0} \dots x _ {n} $
 +
are pairwise distinct interpolation points lying in some interval  $  [ a , b ] $.
  
<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>
+
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
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780047.png" /> is the best approximation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780048.png" /> by algebraic polynomials of degree at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780049.png" />.
+
$$
 +
\| f ( x) - P _ {n} ( x , f  ) \| _ {C [ a , b ] }  \leq  ( 1 +
 +
\lambda _ {n} ) E _ {n} ( f  ) ,
 +
$$
  
For any choice of the interpolation points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780050.png" />, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780051.png" />. For equidistant points a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780052.png" /> exists such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780053.png" />. In case of the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780054.png" />, for points coinciding with the zeros of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780055.png" />-th Chebyshev polynomial, the Lebesgue constants have minimum order of growth, namely
+
where  $  E _ {n} ( f  ) $
 +
is the best approximation of $  f $
 +
by algebraic polynomials of degree at most  $  n $.
  
<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/l05780056.png" /></td> </tr></table>
+
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
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780057.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780058.png" /> times differentiable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780060.png" /> is a given set of numbers ( "approximations of the values fxk" ), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780061.png" /> is the interpolation polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780062.png" /> that takes the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780063.png" /> at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780064.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780065.png" />, and
+
$$
 +
\lambda _ {n}  \approx  \mathop{\rm ln}  n .
 +
$$
  
<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/l05780066.png" /></td> </tr></table>
+
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
 
then
  
<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/l05780067.png" /></td> </tr></table>
+
$$
 +
\| f ^ { ( m) } ( x) - P _ {n}  ^ {(} m) ( x , Y ) \| _ {C [ a , b ] }  \leq
 +
$$
  
<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/l05780068.png" /></td> </tr></table>
+
$$
 +
\leq  \
 +
\| f ^ { ( m) } ( x) - P _ {n}  ^ {(} m) ( x , f  ) \| _ {C [ a , b ] }  +
 +
$$
  
<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/l05780069.png" /></td> </tr></table>
+
$$
 +
+
 +
\lambda _ {nm} \max _ {k = 0 \dots n }  | f ( x _ {k} ) - y _ {k} | .
 +
$$
  
The Lebesgue constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780070.png" /> of an arbitrary interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780071.png" /> are connected with the analogous constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780072.png" /> for the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780073.png" /> by the relation
+
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
  
<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/l05780074.png" /></td> </tr></table>
+
$$
 +
\Lambda _ {nm}  = \left ( b-  
 +
\frac{a}{2}
 +
\right )  ^ {m} \lambda _ {nm} ;
 +
$$
  
in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780075.png" />.
+
in particular, $  \lambda _ {n} = \Lambda _ {n0} $.
  
 
''L.D. Kudryavtsev''
 
''L.D. Kudryavtsev''
  
 
====Comments====
 
====Comments====
The problem to determine  "optimal nodes" , i.e., for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780076.png" /> a fixed positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780077.png" />, to determine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780078.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780079.png" /> is minimal, has been given much attention. S.N. Bernstein [S.N. Bernshtein] (1931) conjectured that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780080.png" /> is minimal when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057800/l05780081.png" /> "equi-oscillates" . Bernstein's conjecture was proved by T.A. Kilgore (cf. [[#References|[a1]]]); historical notes are also included there.
+
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====
 
====References====
 
<table>
 
<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">[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">[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>
+
<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>
 
</table>

Latest revision as of 20:31, 15 November 2023


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.

References

[1] A. Zygmund, "Trigonometric series" , 1 , Cambridge Univ. Press (1988)
[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

$$ \lambda _ {n} = \max _ {a \leq x \leq b } \sum _ { k= 0} ^ { n } | l _ {nk} ( x) | ,\ n = 1 , 2 \dots $$

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. [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