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''of the first kind''
 
''of the first kind''
  
Polynomials that are orthogonal on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021940/c0219401.png" /> with the weight function
+
Polynomials that are orthogonal on the interval $  [ - 1 , 1 ] $
 +
with the weight function
  
<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/c/c021/c021940/c0219402.png" /></td> </tr></table>
+
$$
 +
h _ {1} ( x)  =
 +
\frac{1}{\sqrt {1 - x  ^ {2} }}
 +
,\ \
 +
x \in ( - 1 , 1 ) .
 +
$$
  
 
For the standardized Chebyshev polynomials one has the formula
 
For the standardized Chebyshev polynomials one has the formula
  
<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/c/c021/c021940/c0219403.png" /></td> </tr></table>
+
$$
 +
T _ {n} ( x)  = \cos ( n  \mathop{\rm arc}  \cos  x ) ,\ \
 +
x \in [ - 1 , 1 ] ,
 +
$$
  
 
and the recurrence relation
 
and the recurrence 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/c/c021/c021940/c0219404.png" /></td> </tr></table>
+
$$
 +
T _ {n+1} ( x)  = 2 x T _ {n} ( x) - T _ {n-1} ( x) ,
 +
$$
  
 
by which one can determine the sequence
 
by which one can determine the sequence
  
<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/c/c021/c021940/c0219405.png" /></td> </tr></table>
+
$$
 +
T _ {0} ( x)  = 1 ,\  T _ {1} ( x)  = x ,\ \
 +
T _ {2} ( x)  = 2 x  ^ {2} - 1 ,
 +
$$
  
<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/c/c021/c021940/c0219406.png" /></td> </tr></table>
+
$$
 +
T _ {3} ( x)  = 4 x  ^ {3} - 3 x ,\  T _ {4} ( x)  = 8 x  ^ {4} - 8 x  ^ {2} + 1 ,
 +
$$
  
<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/c/c021/c021940/c0219407.png" /></td> </tr></table>
+
$$
 +
T _ {5} ( x)  = 16 x  ^ {5} - 20 x  ^ {3} + 5 x , .  . . .
 +
$$
  
 
The orthonormalized Chebyshev polynomials are:
 
The orthonormalized Chebyshev polynomials are:
  
<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/c/c021/c021940/c0219408.png" /></td> </tr></table>
+
$$
 +
\widehat{T}  _ {0} ( x)  =
 +
\frac{1}{\sqrt \pi }
 +
T _ {0} ( x)  =
 +
\frac{1}{\sqrt \pi }
 +
,
 +
$$
  
<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/c/c021/c021940/c0219409.png" /></td> </tr></table>
+
$$
 +
\widehat{T}  _ {n} ( x)  = \sqrt {
 +
\frac{2} \pi
 +
} T _ {n} ( x)  = \sqrt {
  
The leading coefficient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021940/c02194010.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021940/c02194011.png" />, is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021940/c02194012.png" />. Hence Chebyshev polynomials with leading coefficient 1 are defined by the formula
+
\frac{2} \pi
 +
}  \cos ( n  \mathop{\rm arc}  \cos  x ) ,\  n \geq  1 .
 +
$$
  
<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/c/c021/c021940/c02194013.png" /></td> </tr></table>
+
The leading coefficient of  $  T _ {n} ( x) $,
 +
for  $  n \geq  1 $,
 +
is  $  2  ^ {n-1} $.  
 +
Hence Chebyshev polynomials with leading coefficient 1 are defined by the formula
  
The zeros of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021940/c02194014.png" />, given by
+
$$
 +
\widetilde{T}  _ {n} ( x)  =
 +
\frac{1}{2  ^ {n- 1} }
 +
T _ {n} ( x)  = \
  
<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/c/c021/c021940/c02194015.png" /></td> </tr></table>
+
\frac{1}{2  ^ {n- 1} }
 +
  \cos ( n  { \mathop{\rm arc}  \cos }  x ) ,\ \
 +
n \geq  1 .
 +
$$
  
frequently occur as interpolation nodes in quadrature formulas. The polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021940/c02194016.png" /> is a solution of the differential equation
+
The zeros of $  T _ {n} ( x) $,
 +
given by
  
<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/c/c021/c021940/c02194017.png" /></td> </tr></table>
+
$$
 +
x _ {k}  ^ {( n)}  = \cos 
 +
\frac{2 k - 1 }{2n}
 +
\pi ,\ \
 +
k = 1 \dots n ,
 +
$$
  
The polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021940/c02194018.png" /> deviate as least as possible from zero on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021940/c02194019.png" />, that is, for any other polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021940/c02194020.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021940/c02194021.png" /> with leading coefficient 1 one has the following condition
+
frequently occur as interpolation nodes in quadrature formulas. The polynomial  $  T _ {n} ( x) $
 +
is a solution of the differential equation
  
<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/c/c021/c021940/c02194022.png" /></td> </tr></table>
+
$$
 +
( 1 - x  ^ {2} ) y  ^ {\prime\prime} - x y  ^  \prime  +
 +
n  ^ {2} y  = 0 .
 +
$$
  
On the other hand, for any polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021940/c02194023.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021940/c02194024.png" /> or less and satisfying
+
The polynomials  $  \widetilde{T}  _ {n} ( x) $
 +
deviate as least as possible from zero on the interval  $  [ - 1 , 1 ] $,
 +
that is, for any other polynomial $  \widetilde{F}  _ {n} ( x) $
 +
of degree $  n $
 +
with leading coefficient 1 one has the following condition
  
<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/c/c021/c021940/c02194025.png" /></td> </tr></table>
+
$$
 +
\max _ {x \in [ - 1 , 1 ] }  | \widetilde{F}  _ {n} ( x) |  > \
 +
\max _ {x \in [ - 1 , 1 ] }  | \widetilde{T}  _ {n} ( x) |
 +
=
 +
\frac{1}{2  ^ {n- 1} }
 +
.
 +
$$
  
one has, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021940/c02194026.png" />, the inequality
+
On the other hand, for any polynomial  $  Q _ {n} ( x) $
 +
of degree  $  n $
 +
or less and satisfying
  
<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/c/c021/c021940/c02194027.png" /></td> </tr></table>
+
$$
 +
\max _ {x \in [ - 1 , 1 ] } \
 +
| Q _ {n} ( x) |  = 1 ,
 +
$$
  
If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021940/c02194028.png" /> is continuous on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021940/c02194029.png" /> and if its modulus of continuity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021940/c02194030.png" /> satisfies the Dini condition
+
one has, for any  $  x _ {0} \in ( - \infty , - 1 ) \cup ( 1 , \infty ) $,
 +
the inequality
  
<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/c/c021/c021940/c02194031.png" /></td> </tr></table>
+
$$
 +
| Q ( x _ {0} ) |  \leq  | T _ {n} ( x _ {0} ) | .
 +
$$
 +
 
 +
If a function  $  f $
 +
is continuous on the interval  $  [ - 1 , 1 ] $
 +
and if its modulus of continuity  $  \omega ( \delta , f  ) $
 +
satisfies the Dini condition
 +
 
 +
$$
 +
\lim\limits _ {\delta \rightarrow 0 }  \omega ( \delta , f ) \
 +
\mathop{\rm ln} 
 +
\frac{1} \delta
 +
  = 0 ,
 +
$$
  
 
then this function can be expanded in a Fourier–Chebyshev series,
 
then this function can be expanded in a Fourier–Chebyshev series,
  
<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/c/c021/c021940/c02194032.png" /></td> </tr></table>
+
$$
 +
f ( x)  = \sum _{n=0} ^  \infty  a _ {n} \widehat{T}  _ {n} ( x) ,\ \
 +
x \in [ - 1 , 1 ] ,
 +
$$
  
which converges uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021940/c02194033.png" />. The coefficients in this series are defined by the formula
+
which converges uniformly on $  [ - 1 , 1 ] $.  
 +
The coefficients in this series are defined by the formula
  
<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/c/c021/c021940/c02194034.png" /></td> </tr></table>
+
$$
 +
a _ {n}  = \int\limits _ { - 1} ^ { 1 }  f ( t) \widehat{T}  _ {n} ( t)
 +
 +
\frac{dt}{\sqrt {1- t  ^ {2} } }
 +
.
 +
$$
  
If the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021940/c02194035.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021940/c02194036.png" />-times continuously differentiable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021940/c02194037.png" /> and if its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021940/c02194038.png" />-th derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021940/c02194039.png" /> satisfies a Lipschitz condition of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021940/c02194040.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021940/c02194041.png" />, then one has the inequality
+
If the function $  f $
 +
is $  p $-
 +
times continuously differentiable on $  [ - 1 , 1 ] $
 +
and if its $  p $-
 +
th derivative $  f  ^ {(} p) $
 +
satisfies a Lipschitz condition of order $  \alpha $,  
 +
i.e. $  f  ^ {(} p) \in  \mathop{\rm Lip}  \alpha $,  
 +
then one has the inequality
  
<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/c/c021/c021940/c02194042.png" /></td> </tr></table>
+
$$
 +
\left | f ( x) - \sum _{k=0} ^ { n }  a _ {k} \widehat{T}  _ {k} ( x) \right |
 +
\leq 
 +
\frac{c _ {1}  \mathop{\rm ln}  n }{n ^ {p + \alpha } }
 +
,\ \
 +
x \in [ - 1 , 1 ] ,
 +
$$
  
where the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021940/c02194043.png" /> does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021940/c02194044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021940/c02194045.png" />.
+
where the constant c _ {1} $
 +
does not depend on $  n $
 +
and $  x $.
  
 
Chebyshev polynomials of the second kind are defined by
 
Chebyshev polynomials of the second kind are defined by
  
<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/c/c021/c021940/c02194046.png" /></td> </tr></table>
+
$$
 +
U _ {n} ( x)  =
 +
\frac{1}{n+1} T _ {n+ 1} ^ { \prime } ( x)  = \sin  [ ( n
 +
+ 1 )  { \mathop{\rm arc}  \cos }  x ]
 +
\frac{1}{\sqrt {1 - x  ^ {2} } } .
 +
$$
 +
 
 +
These polynomials are orthogonal on the interval  $  [ - 1 , 1 ] $
 +
with weight function
  
These polynomials are orthogonal on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021940/c02194047.png" /> with weight function
+
$$
 +
h _ {2} ( x)  = \sqrt {1 - x  ^ {2} } ,\ \
 +
x \in [ - 1 , 1 ] .
 +
$$
  
<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/c/c021/c021940/c02194048.png" /></td> </tr></table>
+
For any polynomial  $  \widetilde{Q}  _ {n} ( x) $
 +
with leading coefficient 1 one has the inequality
  
For any polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021940/c02194049.png" /> with leading coefficient 1 one has the inequality
+
$$
  
<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/c/c021/c021940/c02194050.png" /></td> </tr></table>
+
\frac{1}{2  ^ {n-} 1 }
 +
  = \int\limits _ { - 1} ^ { 1 }
 +
| \widetilde{U}  _ {n} ( x) |  dx  \leq  \int\limits _ { - 1} ^ { 1 }
 +
| \widetilde{Q}  _ {n} ( x) |  dx .
 +
$$
  
 
The Chebyshev polynomials were introduced in 1854 by P.L. Chebyshev (cf. [[#References|[1]]]). Both systems of Chebyshev polynomials are special cases of [[Ultraspherical polynomials|ultraspherical polynomials]] and [[Jacobi polynomials|Jacobi polynomials]].
 
The Chebyshev polynomials were introduced in 1854 by P.L. Chebyshev (cf. [[#References|[1]]]). Both systems of Chebyshev polynomials are special cases of [[Ultraspherical polynomials|ultraspherical polynomials]] and [[Jacobi polynomials|Jacobi polynomials]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.L. Chebyshev,  , ''Collected works'' , '''2''' , Moscow-Leningrad  (1947)  pp. 23–51  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Szegö,  "Orthogonal polynomials" , Amer. Math. Soc.  (1975)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  P.L. Chebyshev,  , ''Collected works'' , '''2''' , Moscow-Leningrad  (1947)  pp. 23–51  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Szegö,  "Orthogonal polynomials" , Amer. Math. Soc.  (1975)</TD></TR>
 +
</table>

Latest revision as of 16:20, 6 January 2024


of the first kind

Polynomials that are orthogonal on the interval $ [ - 1 , 1 ] $ with the weight function

$$ h _ {1} ( x) = \frac{1}{\sqrt {1 - x ^ {2} }} ,\ \ x \in ( - 1 , 1 ) . $$

For the standardized Chebyshev polynomials one has the formula

$$ T _ {n} ( x) = \cos ( n \mathop{\rm arc} \cos x ) ,\ \ x \in [ - 1 , 1 ] , $$

and the recurrence relation

$$ T _ {n+1} ( x) = 2 x T _ {n} ( x) - T _ {n-1} ( x) , $$

by which one can determine the sequence

$$ T _ {0} ( x) = 1 ,\ T _ {1} ( x) = x ,\ \ T _ {2} ( x) = 2 x ^ {2} - 1 , $$

$$ T _ {3} ( x) = 4 x ^ {3} - 3 x ,\ T _ {4} ( x) = 8 x ^ {4} - 8 x ^ {2} + 1 , $$

$$ T _ {5} ( x) = 16 x ^ {5} - 20 x ^ {3} + 5 x , . . . . $$

The orthonormalized Chebyshev polynomials are:

$$ \widehat{T} _ {0} ( x) = \frac{1}{\sqrt \pi } T _ {0} ( x) = \frac{1}{\sqrt \pi } , $$

$$ \widehat{T} _ {n} ( x) = \sqrt { \frac{2} \pi } T _ {n} ( x) = \sqrt { \frac{2} \pi } \cos ( n \mathop{\rm arc} \cos x ) ,\ n \geq 1 . $$

The leading coefficient of $ T _ {n} ( x) $, for $ n \geq 1 $, is $ 2 ^ {n-1} $. Hence Chebyshev polynomials with leading coefficient 1 are defined by the formula

$$ \widetilde{T} _ {n} ( x) = \frac{1}{2 ^ {n- 1} } T _ {n} ( x) = \ \frac{1}{2 ^ {n- 1} } \cos ( n { \mathop{\rm arc} \cos } x ) ,\ \ n \geq 1 . $$

The zeros of $ T _ {n} ( x) $, given by

$$ x _ {k} ^ {( n)} = \cos \frac{2 k - 1 }{2n} \pi ,\ \ k = 1 \dots n , $$

frequently occur as interpolation nodes in quadrature formulas. The polynomial $ T _ {n} ( x) $ is a solution of the differential equation

$$ ( 1 - x ^ {2} ) y ^ {\prime\prime} - x y ^ \prime + n ^ {2} y = 0 . $$

The polynomials $ \widetilde{T} _ {n} ( x) $ deviate as least as possible from zero on the interval $ [ - 1 , 1 ] $, that is, for any other polynomial $ \widetilde{F} _ {n} ( x) $ of degree $ n $ with leading coefficient 1 one has the following condition

$$ \max _ {x \in [ - 1 , 1 ] } | \widetilde{F} _ {n} ( x) | > \ \max _ {x \in [ - 1 , 1 ] } | \widetilde{T} _ {n} ( x) | = \frac{1}{2 ^ {n- 1} } . $$

On the other hand, for any polynomial $ Q _ {n} ( x) $ of degree $ n $ or less and satisfying

$$ \max _ {x \in [ - 1 , 1 ] } \ | Q _ {n} ( x) | = 1 , $$

one has, for any $ x _ {0} \in ( - \infty , - 1 ) \cup ( 1 , \infty ) $, the inequality

$$ | Q ( x _ {0} ) | \leq | T _ {n} ( x _ {0} ) | . $$

If a function $ f $ is continuous on the interval $ [ - 1 , 1 ] $ and if its modulus of continuity $ \omega ( \delta , f ) $ satisfies the Dini condition

$$ \lim\limits _ {\delta \rightarrow 0 } \omega ( \delta , f ) \ \mathop{\rm ln} \frac{1} \delta = 0 , $$

then this function can be expanded in a Fourier–Chebyshev series,

$$ f ( x) = \sum _{n=0} ^ \infty a _ {n} \widehat{T} _ {n} ( x) ,\ \ x \in [ - 1 , 1 ] , $$

which converges uniformly on $ [ - 1 , 1 ] $. The coefficients in this series are defined by the formula

$$ a _ {n} = \int\limits _ { - 1} ^ { 1 } f ( t) \widehat{T} _ {n} ( t) \frac{dt}{\sqrt {1- t ^ {2} } } . $$

If the function $ f $ is $ p $- times continuously differentiable on $ [ - 1 , 1 ] $ and if its $ p $- th derivative $ f ^ {(} p) $ satisfies a Lipschitz condition of order $ \alpha $, i.e. $ f ^ {(} p) \in \mathop{\rm Lip} \alpha $, then one has the inequality

$$ \left | f ( x) - \sum _{k=0} ^ { n } a _ {k} \widehat{T} _ {k} ( x) \right | \leq \frac{c _ {1} \mathop{\rm ln} n }{n ^ {p + \alpha } } ,\ \ x \in [ - 1 , 1 ] , $$

where the constant $ c _ {1} $ does not depend on $ n $ and $ x $.

Chebyshev polynomials of the second kind are defined by

$$ U _ {n} ( x) = \frac{1}{n+1} T _ {n+ 1} ^ { \prime } ( x) = \sin [ ( n + 1 ) { \mathop{\rm arc} \cos } x ] \frac{1}{\sqrt {1 - x ^ {2} } } . $$

These polynomials are orthogonal on the interval $ [ - 1 , 1 ] $ with weight function

$$ h _ {2} ( x) = \sqrt {1 - x ^ {2} } ,\ \ x \in [ - 1 , 1 ] . $$

For any polynomial $ \widetilde{Q} _ {n} ( x) $ with leading coefficient 1 one has the inequality

$$ \frac{1}{2 ^ {n-} 1 } = \int\limits _ { - 1} ^ { 1 } | \widetilde{U} _ {n} ( x) | dx \leq \int\limits _ { - 1} ^ { 1 } | \widetilde{Q} _ {n} ( x) | dx . $$

The Chebyshev polynomials were introduced in 1854 by P.L. Chebyshev (cf. [1]). Both systems of Chebyshev polynomials are special cases of ultraspherical polynomials and Jacobi polynomials.

References

[1] P.L. Chebyshev, , Collected works , 2 , Moscow-Leningrad (1947) pp. 23–51 (In Russian)
[2] G. Szegö, "Orthogonal polynomials" , Amer. Math. Soc. (1975)
How to Cite This Entry:
Chebyshev polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chebyshev_polynomials&oldid=16283
This article was adapted from an original article by P.K. Suetin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article